Journal of Lie Theory Volume 14 (2004) 443–479 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

Hierarchy of Closures of Matrix Pencils

Dmitri D. Pervouchine

Communicated by K. H. Hofmann

Abstract. The focus of this paper is the standard linear representation of the group SL_{n}(ℂ)×SL_{m}(ℂ)×SL_{2}(ℂ), that is, the tensor product of the corresponding tautological representations. Classification of its orbits is a classic problem, which goes back to the works of Kronecker and Weierstrass. Here, we summarize some known results about standard linear representations of SL_{n}(ℂ) × SL_{m}(ℂ) × SL_{2}(ℂ), GL_{n}(ℂ) × GL_{m}(ℂ), SL_{n}(ℂ) × SL_{m}(ℂ), and GL_{n}(ℂ) × GL_{m}(ℂ) × GL_{2}(ℂ), classify the orbits and describe their degenerations. For the case n ≠ m, we prove that the algebras of invariants of the actions of SL_{n}(ℂ) × SL_{m}(ℂ) × SL_{2}(ℂ) and SL_{n}(ℂ) × SL_{m}(ℂ) are generated by one polynomial of degree nm/|n − m|, if d = |n − m| divides n (or m), and are trivial otherwise. It is also shown that the null cone of SL_{n}(ℂ) × SL_{m}(ℂ) × SL_{2}(ℂ) is irreducible and contains an open orbit if n ≠ m. We relate the degenerations of orbits of matrix pencils to the degenerations of matrix pencil bundles and prove that the closure of a matrix pencil bundle consists of closures of the corresponding orbits and closures of their amalgams. From this fact we derive the degenerations of orbits of the four groups listed above. All degenerations are cofined to the list of minimal degenerations, which are summarized as transformations of Ferrer diagrams. We tabulate the orbits of matrix pencils up to seventh order and portray the hierarchy of closures of 2 × 2, 3 × 3, 4 × 4, 5 × 5, 5 × 6 and 6 × 6 matrix pencil bundles.

1. Introduction

Although classification of orbits of the standard linear representation of SL_{n_{1}}(ℂ) × ⋯ × SL_{n_{s}}(ℂ) is trivial for s ≤ 2, it is no longer feasible in any reasonable sense even for s = 3. The simplest nontrivial case of this generic problem is when one of the tensor factors is SL_{2}(ℂ). Here, we investigate this case, the standard linear representation of SL_{n,m,2} = SL_{n}(ℂ) × SL_{m}(ℂ) × SL_{2}(ℂ) in ℂ^{n,m,2} = ℂ^{n} ⊗ ℂ^{m} ⊗ ℂ^{2}, and describe its orbits and their degenerations.

The space ℂ^{n,m,2} is endowed with the natural actions of three other groups: GL_{n}(ℂ) × GL_{m}(ℂ), SL_{n}(ℂ) × SL_{m}(ℂ), and GL_{n}(ℂ) × GL_{m}(ℂ) × GL_{2}(ℂ). They are denoted by GL_{n,m}, SL_{n,m}, and GL_{n,m,2}, respectively. If bases in ℂ^{n}, ℂ^{m}, and ℂ^{2} are chosen then the components T^{ijk} of a tensor T ∈ ℂ^{n,m,2} form two n × m matrices, whose entries are T^{ij1} and T^{ij2}, respectively. An element of ℂ^{n,m,2} can

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag be regarded as a pair of complex n × m matrices A and B. Then it is called the matrix pencil and is denoted by A + λB, where λ is a varying coefficient. If n = m then the matrix pencil is called square; otherwise it is called rectangular. The actions of GL_{n,m,2} and SL_{n,m,2} on matrix pencils are given by formula

(P, Q, R)∘(A+λB) = (r_{11}P AQ^{−1} +r_{12}P BQ^{−1})+λ(r_{21}P AQ^{−1} +r_{22}P BQ^{−1}), (1)

where r_{ij} are the entries of R^{−1}. Two matrix pencils are said to be G-equivalent if one of them is mapped to the other by a transformation that belongs to G, where G is one of the groups listed above. Two matrix pencils are said to be equivalent (no prefix used), if they are GL_{n,m}-equivalent.

Description of equivalence classes of matrix pencils under the action of GL_{n,m} was obtained by Weierstrass and Kronecker [15, 25]. Here we state some of their results. The direct sum of a n_{1} × m_{1} matrix pencil P_{1} = X_{1} + λY_{1} and a n_{2} × m_{2} matrix pencil P_{2} = X_{2} + λY_{2} is the (n_{1} + n_{2}) × (m_{1} + m_{2}) matrix pencil P_{1} ⊕ P_{2} = (X_{1} ⊕ X_{2}) + λ(Y_{1} ⊕ Y_{2}), where Z_{1} ⊕ Z_{2} denotes the diagonal block matrix composed of Z_{1} and Z_{2}. A matrix pencil is said to be indecomposable if it cannot be represented as a direct sum of two non-trivial matrix pencils. We also consider n × 0 and 0 × m matrix pencils; by this we mean that if such pencil is present in a direct sum then the corresponding matrices are given rows or columns of zeroes. Every indecomposable matrix pencil is equivalent to one of the following matrix pencils:

L_{k} = \underbrace{(\begin{matrix}1 & λ \\ & 1 & λ \\ & & ... & ... \\ & & & 1 & λ \end{matrix})}_{k+1}, R_{k} = (\begin{matrix}1 \\ λ & 1 \\ & λ & ... \\ & & ... & 1 \\ & & & λ \end{matrix})\} k + 1, k ≥ 0,

D_{k}(μ) = E_{k} + λJ_{k}(μ), k > 0,
D_{k}(∞) = λJ_{k}(1), k > 0,

where E_{k} is the k-th order identity matrix, and J_{k}(μ) is the k-th order Jordan matrix with eigenvalue μ. Every matrix pencil is equivalent to the direct sum

L_{k_{1}} ⊕ ⋯ ⊕ L_{k_{p}} ⊕ R_{l_{1}} ⊕ ⋯ ⊕ R_{l_{q}} ⊕ D_{n_{1}}(μ_{1}) ⊕ ⋯ ⊕ D_{n_{s}}(μ_{s}),  (2)

where k_{1}, …, k_{p} and l_{1}, …, l_{q} are the minimal indices of rows and columns, respectively, and μ_{1}, …, μ_{s} are the eigenvalues of the matrix pencil. The decomposition (2) is called the Kronecker canonical form of a matrix pencil. The set of indecomposable matrix pencils in (2) is defined unambiguously up to a transposition. A matrix pencil A + λB is said to be regular, if it is square and det(A + λB) is not identically zero. Otherwise, it is called singular. A matrix pencil is said to be completely singular, if its Kronecker canonical form has no regular blocks. Every matrix pencil P is decomposed into sum of regular and completely singular matrix pencils P^{reg} and P^{sing}. They are called the regular part and the singular part of the matrix pencil, respectively. A matrix pencil is said to be perfect, if its Kronecker canonical form is L_{k} ⊕ ⋯ ⊕ L_{k} or R_{k} ⊕ ⋯ ⊕ R_{k}. Otherwise, it is said to be imperfect.

Ja’ja extended the results of Kronecker and Weierstrass to the action of GL_{n,m,2} [13]. In particular, he proved that completely singular matrix pencils are GL_{n,m,2}-equivalent if and only if they are GL_{n,m}-equivalent. Thus, the classification of matrix pencils under the action of GL_{n,m,2} consists of two separate problems: classification of completely singular matrix pencils under the action of GL_{n,m}, and classification of regular matrix pencils with respect to the action of GL_{n,m,2}.

It took hundred years since the works of Kronecker and Weierstrass to get the description of closures of GL_{n,m}-orbits. During this time, investigations in the theory of matrix pencils were motivated mainly by applications to differential equations and linear-quadratic optimization problems [23, 8]. Several algoritms for calculation of eigenvalues and Kronecker canonical form of matrix pencils were designed [10, 24, 2]. However, as in the case of calculation of the Jordan normal form of a matrix, these algorithms have a common problem: they cannot distinguish between close points that belong to different orbits. The description of closures of GL_{n,m}-orbits was obtained in 1986 by Pokrzywa [21] and later by Bongartz [3]. Pokrzywa has found and systematically described the main types of degenerations of orbits and has shown that the other degenerations are the combinations of those main ones. Bongartz has got his result by using the theory of representations of quivers. However, as the number of orbits is infinite, it is more convenient to describe the closures of bundles of matrix pencils. This has been done by Edelman et al [6]. In §7. we give a careful proof for the criterion that has been stated in their work.

Classification of orbits is closely related to description of invariants. A pair of matrices A and B that are defined up to the action of GL_{n,m} is a class of equivalent representations of the quiver of type Ã_{1}. In fact, description of invariants of SL_{n,m} for arbitrary n and m can be derived from the general results on the invariants of representations of tame quivers. For rectangular matrix pencils GL_{n,m} the description of SL_{n,m}-invariants can be obtained from [11]. In case of square matrix pencils the coefficients of the binary form det(λA + μB) coincide with the semi-invariants constructed in [22], which generate the algebra of SL_{n,m}- invariants according to [12, theorem 2.3]. Hence, the algebra of invariants of the standard linear representation of SL_{n,n,2} is isomorphic to the algebra of invariants of binary forms of degree n.

Thus, the works related to the standard linear representations of GL_{n,m}, SL_{n,m}, GL_{n,m,2}, and SL_{n,m,2}, can be summarized in the following table (by asterics we denote the questions addressed in this work):

Group Classification of orbits Orbits’ closures Invariants GL^{n,m}  [15, 25]  [21], [3], [6]  trivial GL^{n,m,2}  [13]  ⋆  trivial SL^{n,m}  ⋆  ⋆  [22], [12], [16] SL^{n,m,2}  ⋆  ⋆  ⋆ .

For several small n and m the standard linear representation of SL_{n}(ℂ)×SL_{m}(ℂ)× SL_{2}(ℂ) has good properties, which make it more tractable than in generic case. For instance, if n = m = 4 then the representation is visible, that is, the number of nilpotent orbits is finite [14]. This case was extensively studied in [19, 20]. Other “exceptional cases” were investigated in works of Ehrenborg and Nurmiev on 2 × 2 × 2 and 3 × 3 × 3 matrices [7, 17]. In regard of visible representations it is also worth of mentioning the work of Parfenov [18] and its generalization to the case of real matrices that was obtained recently by Dokovic and Tingley [5].

Now we ouline the structure of the paper. We classify the orbits of GL_{n,m}, SL_{n,m}, GL_{n,m,2}, and SL_{n,m,2} separately for regular (section 2.), imperfect singular (section 3.), and perfect (section 4.) matrix pencils. In section 5. we summarize the results of the classification. In section 4. we also prove that the algebra of invariants of rectangular n × m matrix pencils is generated by one polynomial of degree nm/d, if d = |n−m| divides n and m and is trivial otherwise. In section 6. we introduce a notion of matrix pencil bundle, which is a union of orbits of GL_{n,m} over all possible eigenvalues given that the sets of minimal indices of rows, minimal indices of columns, and multiplicites of eigenvalues are fixed. When we take limits of elements of a matrix pencil bundle some of the eigenvalues may coalesce; this process is called amalgamation. In section 7. we prove that the closure of a matrix pencil bundle consists of closures of the corresponding orbits and closures of their amalgams (theorem 7.5). In sections 8. and 9. we study closures of orbits of GL_{n,m,2} and SL_{n,m,2}. In order to describe the hierarchy of closures (also called the Hasse diagram), we come up with a list of minimal degenerations, which are summarized in section 10. as a set of rules for transformations of Ferrer diagrams. In section 11. we focus on the geometry of the null cones of SL_{n,m} and SL_{n,m,2}. In particular, if n ≠ m then the null cone of SL_{n,m,2} is irreducible and contains an open orbit. If n = m then the null cone of SL_{n,m,2} is also irreducible but does not contain an open orbit for n > 4. In section 12. we tabulate the orbits of matrix pencils (up to seventh order) and produce figures for the hierarchy of closures of 2 × 2, 3 × 3, 4 × 4, 5 × 5, 5 × 6 and 6 × 6 matrix pencil bundles.

In what follows, the base field is the field ℂ of complex numbers. If V is a vector space over ℂ then V^{∗}, L(V), L_{0}(V), Λ^{k}V , S^{k}V , and ⊗^{k}V denote the dual vector space, the space of all linear endomorphisms of V , the space of linear endomorphisms of V with zero trace, the k-th exterior, symmetric, and tensor powers of V , respectively. The space Λ^{k}V is identified with the subspace of ⊗^{k}V in a natural way. If e_{1}, …, e_{n} is a base in V then e^{1}, …, e^{n} denotes the dual base of V^{∗}. The components of tensors T ∈ ⊗^{k}V and T^{∗} ∈ ⊗^{k}V^{∗} are denoted by T^{i_{1}...i_{k}} and T_{i_{1}...i_{k}}^{∗}, respectively. The mapping π: Λ^{k}V →(Λ^{n−k}V)^{∗} given by

(πv^{i_{1}...i_{k}})(q^{i_{k+1}...i_{n}}) = det_{i_{1}...i_{k}i_{k+1}...i_{n}}v^{i_{1}...i_{k}}q^{i_{k+1}...i_{n}},  (3)

where det(e_{1}, …, e_{n}) = 1, is an isomorphism between Λ^{k}V and (Λ^{n−k}V)^{∗}.

Let G be a group acting on a set X. Denote by Gx (or O(x), if choice of G is clear from the context) the orbit of x ∈ X under the action of G. If X is a vector space over ℂ, and G is an algebraic group then ℂ[X] and ℂ[X]^{G} denote the algebra of polynomial functions on X and its subalgebra of G-invariant polynomial functions, respectively. The spectrum of ℂ[X]^{G} is denoted by X//G. An element x ∈ X is called nilpotent if the closure of its orbit contains zero element. The set of all nilpotent elements of X is called the null cone and is denoted by 𝔑_{X}. The dimension and the codimension of Y ⊂ X are denoted by dim Y and cod(X, Y), respectively. The closure of Y in X (in the topology of ℂ) is denoted by \overline{Y} . The number of elements of a finite set S is denoted by |S|.

2. Regular matrix pencils

Regular matrix pencils are square. Every regular matrix pencil is SL_{n,n}- equivalent to the matrix pencil c · P, where P is the Kronecker canoical form. Obviously, c_{1} · P and c_{2} · P are SL_{n,n}-equivalent if and only if c_{1}^{n} = c_{2}^{n}, where n is the order of P. Every regular matrix pencil is SL_{n,m,2}-equivalent to E+λD, where E is the identity matrix and D is a Jordan matrix. The matrix pencils E + λD and E + λD′ are GL_{n,m,2}-equivalent if and only if there exist P, Q ∈ GL_{n}(ℂ) and R ∈ GL_{2}(ℂ) such that

\{ \begin{matrix}P (r_{11}E + r_{12}D)Q^{−1} = E \\ P (r_{21}E + r_{22}D)Q^{−1} = D′. \end{matrix}  (4)

Equivalently, there exist P ∈ GL_{n}(ℂ) and R ∈ GL_{2}(ℂ) such that P (φ_{R}(D))P^{−1} = D′. Here φ_{R} is the linear fractional transformation defined by φ_{R}(X) = (r_{21}E + r_{22}X)(r_{11}E + r_{12}X)^{−1}. If we want E + λD and E + λD′ to be SL_{n,m,2}-equivalent then we need to have det(r_{11} + r_{12}D) = det R = 1. It is clear that proportional matrices define equal fractional transformations. Therefore we can assume that R ∈ GL_{2}(ℂ) has the following property:

det(r_{11} + r_{12}D) = (det R)^{n/2}.  (5)

We now state some of our previous results [20]. The matrix pencils E + λD and E + λD′ are GL_{n,m,2}-equivalent (respectively, SL_{n,m,2}-equivalent) if and only if the eigenvalues of D are mapped to the eigenvalues of D′ by a linear fractional transformation (respectively, by a linear fractional transformation that satisfies (5)) that preserves the multiplicities of the corresponding eigenvalues. The polynomial mapping θ: ℂ^{n,n,2} → Y = S^{n}(ℂ^{2}^{∗}), that takes each matrix pencil A + λB to the binary form f(α, β) = det(αA + βB) is dominant and SL_{n,n,2}-equivariant assuming that the first and the second factor of SL_{n,n,2} act trivially on Y . The algebra ℂ[ℂ^{n,n,2}]^{SL^{n,n}} is generated by coefficients of the binary form det(αA + βB). It follows from this theorem that the algebra of SL_{n,n,2}-invariants of n × n matrix pencils is isomorphic to the algebra of SL_{2}(ℂ)-invariants of binary forms of degree n, and a square matrix pencil P is nilpotent under the action of SL_{n,n,2} if and only if the binary form θ(P) is nilpotent.

3. Imperfect singular matrix pencils

It follows from [13, theorem 3] that all perfect matrix pencils are GL_{n,m,2}- equivalent. Imperfect singular matrix pencils are GL_{n,m,2}-equivalent if and only if their regular components are GL_{n,m,2}-equivalent and their singular components are GL_{n,m}-equivalent.

Now we focus on SL_{n,m}- and SL_{n,m,2}-equivalence of imperfect singular matrix pencils. The following lemma is an analog of [20, lemma 2] for rectangular matrix pencils.

Lemma 3.1. Let P_{1} and P_{2} be matrix pencils, and let n_{1} × m_{1} and n_{2} × m_{2} be the sizes of P_{1} and P_{2}, respectively. If the pairs (n_{1}, m_{1}) and (n_{2}, m_{2}) are not proportional then P_{1} ⊕ P_{2} and (α · P_{1}) ⊕ (β · P_{2}) are SL_{n,m}-equivalent for any pair of non-zero complex numbers α and β.

Proof. Put P = diag(α_{1}E_{n_{1}}, β_{1}E_{n_{2}}) and Q = diag(α_{2}E_{m_{1}}, β_{2}E_{m_{2}}), where E_{k} is the k-th order identity matrix. Then P_{1}⊕P_{2} and (α·P_{1})⊕(β·P_{2}) are equivalent under the action of (P, Q) ∈ SL_{n,m} if and only if α_{1}^{n_{1}}β_{1}^{n_{2}} = 1, α_{2}^{m_{1}}β_{2}^{m_{2}} = 1, α_{1}α_{2}^{−1} = α and β_{1}β_{2}^{−1} = β. Obviously, these equations have a common solution if the pairs (n_{1}, m_{1}) and (n_{2}, m_{2}) are not proportional.

Corollary 3.2. The following matrix pencils

(i) P_{1} = L_{p} ⊕ R_{q} and P_{2} = (α · L_{p}) ⊕ (β · R_{q}),

(ii) P_{1} = L_{p} ⊕ L_{q} and P_{2} = (α · L_{p}) ⊕ (β · L_{q}) for p ≠ q,

(iii) P_{1} = R_{p} ⊕ R_{q} and P_{2} = (α · R_{p}) ⊕ (β · R_{q}) for p ≠ q,

(iv) P_{1} = L_{p} ⊕ Q and P_{2} = (α · L_{p}) ⊕ (β · Q),

(v) P_{1} = R_{p} ⊕ Q and P_{2} = (α · R_{p}) ⊕ (β · Q) are SL_{n,m}-equivalent for any square l ×l matrix pencil Q and any pair of non-zero complex numbers α and β.

Theorem 3.3. Imperfect singular matrix pencils are SL_{n,m}-equivalent if and only if they are GL_{n,m}-equivalent.

Proof. Suppose that P_{1} and P_{2} are equivalent. Then P_{1} and k ·P_{2} are SL_{n,m}- equivalent for some k ∈ ℂ. We may replace P_{1} and P_{2} with their Kronecker canonical forms. If P_{2} is a square singular matrix pencil then P_{2} = L_{p} ⊕ R_{q} ⊕ Q. It is clear that k · P_{2} and (k′ · L_{p}) ⊕ (k′ · R_{q}) ⊕ Q are equivalent under the action of SL_{n,m}. Then P_{1} and P_{2} are SL_{n,m}-equivalent by corollary 3.2. If P_{2} is an imperfect rectangular matrix pencil then its Kronecker canonical form contains two left blocks of different sizes or two right blocks of different sizes or a non-trivial regular block. It follows from corollary 3.2 that P_{1} and P_{2} are SL_{n,m}-equivalent. The proof of the converse is trivial.

Theorem 3.4. Imperfect singular matrix pencils are SL_{n,m,2}-equivalent if and only if they are GL_{n,m,2}-equivalent.

Proof. Suppose that P_{1} and P_{2} are GL_{n,m,2}-equivalent. By [13, theorem 3], their singular components are GL_{n,m}-equivalent, and their regular components are SL_{n,m,2}-equivalent up to a multiplicative factor. Therefore, P_{1} is SL_{n,m,2}-equivalent to P_{1}^{sing} ⊕ (k · P_{2}^{reg}) that is, in turn, SL_{n,m}-equivalent to (k′ · P_{1}^{sing}) ⊕ P_{2}^{reg}. Then P_{1} and P_{2} are SL_{n,m,2}-equivalent by previous theorem.

4. Perfect matrix pencils

Without loss of generality we consider only perfect n × m matrix pencils for m > n. Then d = m − n divides n and every perfect matrix pencil is SL_{n,m}- equivalent to L_{k} ⊕ ⋯ ⊕ L_{k} up to a multiplicative factor. Here L_{k} ⊕ ⋯ ⊕ L_{k} denotes the sum of d indecomposable k × l matrix pencils, where k = n/d and l = m/d = k+1. The matrix pencil L_{k} ⊕⋯⊕L_{k} is SL_{n,m}-equivalent to the matrix pencil P = [E_{n}|0_{nd}] + λ[0_{nd}|E_{n}], where E_{n} is the n-th order identity matrix, 0_{nd} is the zero n × d matrix, and [A|B] denotes the matrix obtained by attachment of the matrix B to the right of the matrix A. The following lemma shows that the statement of theorem 3.3 is correct only for imperfect matrix pencils, that is, it is impossible to multiply a perfect matrix pencil by an arbitrary complex number by a transformation from SL_{n,m}.

Theorem 4.1. Let P be a matrix pencil. Then P and c · P are SL_{n,m}- equivalent iff c^{km} = 1.

Proof. Assume that P = [E_{n}|0_{nd}] + λ[0_{nd}|E_{n}]. Let matrices P ∈ SL_{n}(ℂ) and Q ∈ SL_{m}(ℂ) be such that P P Q^{−1} = c · P. Consider the following partition of the matrix Q

(\begin{matrix}Q′ & Q_{1} \\ Q_{2} & Q_{0} \end{matrix}) ,

where Q′ is a square n × n matrix, Q_{1}, Q_{2}, and Q_{0} are the corresponding complementary matrices. Then P [E|0] Q^{−1} = [E|0] implies P = Q′ and Q_{1} = 0. Applying similar arguments to P [0|E] Q^{−1} = [0|E] we get that Q is composed of square d × d-blocks

(\begin{matrix}Q_{11} & Q_{12} & … & Q_{1k} & 0 \\ 0 & ⋮ & & ⋮ & 0 \\ 0 & Q_{l2} & … & Q_{lk} & Q_{ll} \end{matrix}) ,

and

P = (\begin{matrix}Q_{11} & Q_{12} & … & Q_{1k} \\ 0 & ⋮ & & ⋮ \\ 0 & Q_{k2} & … & Q_{kk} \end{matrix}) = (\begin{matrix}Q_{22} & … & Q_{2k} & 0 \\ ⋮ & & ⋮ & 0 \\ Q_{l2} & … & Q_{lk} & Q_{ll} \end{matrix}) .

Therefore, Q_{32} = ⋯ = Q_{l2} = 0, Q_{1k} = ⋯ = Q_{k−1 k} = 0 and Q_{11} = ⋯ = Q_{ll}. Iterating this process k times we get that P = diag(A, …, A) and Q = diag(cA, …, cA), where A is a d × d matrix. Therefore, (det A)^{k} = 1 and c^{m}(det A)^{k+1} = 1. This completes the proof.

It follows from theorem 4.1 that SL_{n,m} has infinite number of orbits in ℂ^{n,m,2}. This fact indicates that the algebra of invariants is non-trivial. It turns out that ℂ[ℂ^{n,m,2}]^{SL^{n,m}} has only one generator, which is also SL_{n,m,2}-invariant. For simplicity we now assume that d = 1. Then, n = k and m = k + 1. Denote by Δ_{i} the n-th order minor of the matrix A + λB that doesn’t contain the i-th column; they are polynomial functions of λ

Δ_{i}(λ) = ∑_{s=0}^{n} Δ_{is}λ^{n−s},

and their coefficients Δ_{is} are n-th degree polynomials of the entries of A and B. The numbers Δ_{is} form a square (n + 1) × (n + 1) matrix Δ, whose determinant we denote by ω(P).

Lemma 4.2. The polynomial ω is invariant under the action of SL_{n,m,2}.

Proof. It is clear that ω(P) is invariant under the left action of SL_{n}(ℂ). Consider the matrix A + λB as a collection of columns a_{1}(λ), …, a_{m}(λ). If we add the j-th column multiplied by some factor c to the i-th column then Δ_{j} is converted to Δ_{j} + c(−1)^{j−i}Δ_{i} and the other minors don’t change. This proves that det(Δ) is invariant under the right action of SL_{m}(ℂ). Now consider the transformation A ↦ A+cB. It corresponds to the transformation λ ↦ λ+c of the varying coefficient, which adds a linear combination of the successive columns to each column of Δ. Transposition of A and B, being combined with simultaneous multiplication of B by −1, converts Δ_{i}(λ) to λ^{n}Δ_{i}(−λ^{−1}). These transformations don’t affect det(Δ). Thus, ω(P) is SL_{n,m,2}-invariant.

In order to construct the invariant for an arbitrary d we will use the values of Δ_{i}(λ) instead of their coefficients. Let e_{1}, …, e_{m} be a base in ℂ^{m}, and let e^{1}, …, e^{m} be the corresponding dual base. Denote the tautological linear representation of SL_{p}(ℂ) in ℂ^{p} by ρ_{p} and consider the morphism η of the linear representations ρ_{n} ⊗ ρ_{m} and Λ^{n}ρ_{m}^{∗}, that takes each matrix A to the n-vector

η(A) = ∑_{i_{1}<···<i_{n}} Δ_{i_{1}...i_{n}}(A) e^{i_{1}} ∧ ⋯ ∧ e^{i_{n}}.

Here Δ_{i_{1}...i_{n}}(A) is the n-th order minor of the matrix A that contains the columns i_{1}, …, i_{n}. One can show that the right multiplication of A by a unimodular matrix Q corresponds to the left multiplication of η(A) by Q^{⊤}. Obviously, the left multiplication of A by a unimodular matrix doesn’t affect the n-th order minors. Therefore, η is SL_{n,m}-invariant (we assume that SL_{n}(ℂ) acts on (Λ^{n}ℂ^{m})^{∗} trivially). Recall that (Λ^{n}ℂ^{m})^{∗} is identified with Λ^{d}ℂ^{m} by the SL_{n,m}-equivariant morphism (3). Then η can be regarded as the following chain of morphisms: Hom((ℂ^{n})^{∗}, ℂ^{m}) → Hom(Λ^{n}(ℂ^{n})^{∗}, Λ^{n}ℂ^{m}) → Hom(Λ^{n}(ℂ^{m})^{∗}, Λ^{n}ℂ^{n}) = Λ^{d}ℂ^{m}. Now chose some distinct complex numbers λ_{1}, …, λ_{l} and define ω by formula

ω(P) = (−1)^{\frac{d(d−1)}{2} · \frac{l(l−1)}{2}} η(P(λ_{1})) ∧ ⋯ ∧ η(P(λ_{l})).  (6)

The morphism ω takes each matrix pencil P to the element of Λ^{m}ℂ^{m} ≃ ℂ and is invariant under the action of SL_{n,m}.

Theorem 4.3. The morphism ω is invariant under the action of SL_{n,m,2}.

Proof. It is easy to prove that if P = L_{k} ⊕ ⋯ ⊕ L_{k} then ω(P) is equal to Wandermond determinant of λ_{1}, …, λ_{l} and otherwise it is equal to zero. The action of SL_{2}(ℂ) on ℂ^{n,m,2} is generated by transformations A + λB ↦ (A + cB) + λB and A + λB ↦ B − λA. They correspond to the following transformations of λ: P(λ) ↦ P(λ + c) and P(λ) ↦ λP(−λ^{−1}). If we add the same constant to all λ_{1}, …, λ_{l} then the Wandermond determinant will not change. The transformation P(λ) ↦ λP(−λ^{−1}) doesn’t affect ω, too, as it inverses the order of the columns in the Wandermond determinant and multiplies the odd columns by −1. Therefore, ω(P) is SL_{n,m,2}-invariant.

Corollary 4.4. Perfect matrix pencils are SL_{n,m,2}-equivalent if and only if they are SL_{n,m}-equivalent. It follows from this theorem that the set of perfect matrix pencils is an open and dense subset in the set of all n × m matrix pencils, and the orbits of a perfect matrix pencil under the actions of SL_{n,m} and SL_{n,m,2} are closed.

Theorem 4.5. If d = m−n divides n, then ℂ[ℂ^{n,m,2}]^{SL^{n,m}} ≃ ℂ[ℂ^{n,m,2}]^{SL^{n,m,2}} ≃ ℂ[ω]. Otherwise, ℂ[ℂ^{n,m,2}]^{SL^{n,m}} ≃ ℂ[ℂ^{n,m,2}]^{SL^{n,m,2}} ≃ ℂ.

Proof. If d doesn’t divide n then all elements of ℂ^{n,m,2} are imperfect singular matrix pencils. Then by theorems 3.3 and 3.4 all orbits are nilpotent and therefore ℂ[ℂ^{n,m,2}]^{SL^{n,m}} ≃ ℂ[ℂ^{n,m,2}]^{SL^{n,m,2}} ≃ ℂ. Now let d divide n. Consider the onedimensional subspace L spanned over the perfect matrix pencil L_{k} ⊕ ⋯ ⊕ L_{k}. The set SL_{n,m} · L is a Zarisski open set, which is cut by ω(P) ≠ 0. Then L is a Chevallier section, that is, the morphism of restriction of invariants on L is an isomorphism, and therefore the transcendence degree of ℂ[ℂ^{n,m,2}]^{SL^{n,m}} is equal to one. By theorem 4.1, we have |W | = km = deg(ω). Thus, ω is a generator of ℂ[ℂ^{n,m,2}]^{SL^{n,m}} and, since the generic SL_{n,m}-orbits and SL_{n,m,2}-orbits coincide, also a generator of ℂ[ℂ^{n,m,2}]^{SL^{n,m,2}} .

5. Classification of matrix pencils

A partition of an integer n is a non-decresing infinite sequence of integers n = (n_{1}, n_{2}, …) that sums up to n. It is obvious that the number of non-zero terms in such sequence is finite. Let n and m be the partitions of the integers n and m, respectively. The sum n+m is the termwise sum of the sequences of n and m. The union n∪m is the union of terms of n and m sorted in descending order. The conjugate partition n^{∗} is defined by n_{k}^{∗} = |{i | n_{i} ≥ k}|. For any partitions n and m we have (n + m)^{∗} = n^{∗} ∪ m^{∗} and (n^{∗})^{∗} = n. We say that the partition n dominates the partition m and write n ≥ m, if n_{1}+⋯+n_{k} ≥ m_{1}+⋯+m_{k} for all k. If n = (n_{1}, n_{2}, …) and a is an integer then n + a denotes (n_{1} + a, n_{2} + a, …). If n and ñ are the partitions of the same integer n and n ≥ ñ then ñ^{∗} ≥ n^{∗}. The partition m (respectively, n) is said to be the lowering (respectively, the heightening) of the partition n (respectively, m), if n > m, that is, n ≥ m and n ≠ m. The lowering (respectively, the heightening) is said to be minimal, if there is no partition k such that n > k > m. The partitions are usually illustrated by Ferrer diagrams. The i-th column of the Ferrer diagram contains n_{i} cells. The Ferrer diagram of n^{∗} is obtained from the Ferrer diagram of n by transposition. The Ferrer diagram of a minimal lowering of the partition n is obtained from the Ferrer diagram of n by crumbling or deletion of its rightmost cell (if it exists).

Let P be a matrix pencil. Denote the number of blocks L_{k}, R_{k} and D_{k}(μ) in the Kronecker canonical form of P by l_{k}(P), r_{k}(P) and d_{k}(μ, P), respectively. Define l(P) = ∑ l_{k}(P) and r(P) = ∑ r_{k}(P). The number dim U − l(P) = dim V − r(P) is called the normal rank of P and is denoted by nrk(P). Define the partitions 𝔇(μ, P) = (𝔡_{1}(μ, P), 𝔡_{2}(μ, P), …), 𝔏(P) = (𝔩_{0}(P), 𝔩_{1}(P), …), and ℜ(P) = (𝔯_{0}(P), 𝔯_{1}(P), …) by

𝔩_{i}(P) = ∑_{k≥i} l_{k}(P)  (7)
𝔯_{i}(P) = ∑_{k≥i} r_{k}(P)  (8)
𝔡_{i}(μ, P) = ∑_{k≥i} d_{k}(μ, P).  (9)

The sequences l_{i}(P), r_{i}(P) and d_{i}(μ, P) are determined uniquely by partitions 𝔇(μ, P), 𝔏(P) and ℜ(P). Note that the partition 𝔇(μ, P) is conjugate to the set of orders of regular blocks corresponding to the eigenvalue μ. For instance, if 𝔇(μ, P) = (2, 1, 1) and 𝔏(P) = (1, 1) then P = D_{1}(μ) + D_{3}(μ) + L_{1}.

Now the classes of equivalent matrix pencils are described as follows:

1. A class of GL_{n,m}-equivalent matrix pencils is given by the set {μ_{1}, …, μ_{s}} of (distinct) eigenvalues and the partitions 𝔏(P), ℜ(P), and 𝔇(μ_{j}, P).

2. A calss of GL_{n,m,2}-equivalent matrix pencils is given by the set of eigenvalues that are defined up to a linear fractional transformation and the partitions 𝔏(P), ℜ(P), and 𝔇(μ_{j}, P).

3. A class of SL_{n,m}-equivalent matrix pencils is given by

(a) the set of eigenvalues and the partitions 𝔏(P), ℜ(P), and 𝔇(μ_{j}, P), if the matrix pencil is singular and imperfect;

(b) the proportionality factor (between matrix pencil and its Kronecker canonical form) that is defined up to multiplication by km-th root of unity, if the matrix pencil is perfect;

(c) the set of eigenvalues, the partitions 𝔇(μ_{j}, P), and the proportionality factor (as before) that is defined up to multiplication by n-th root of unity, if the matrix pencil is regular;

4. A class of SL_{n,m,2}-equivalent matrix pencils is given by

(a) the set of eigenvalues that are defined up to a linear fractional transformation and the partitions 𝔏(P), ℜ(P), and 𝔇(μ_{j}, P), if the matrix pencil is singular and imperfect;

(b) the proportionality factor that is defined up to multiplication by km-th root of unity, if the matrix pencil is perfect;

(c) the set of eigenvalues that are defined up to a linear fractional transformation, which satisfies (5) and the partitions 𝔇(μ_{j}, P), if the matrix pencil is regular.

Remark 5.1. In all tables and figures we use the following simplified notation of matrix pencils. If 𝔇(μ_{i}, P) = (n_{i1}, n_{i2}, …) then the regular component of P is denoted by J_{n_{1}}(n_{11}, n_{12}, …) ⊕ ⋯ ⊕ J_{n_{s}}(n_{s1}, n_{s2}, …). The integers n_{ij} that are equal to 0 or 1 are omitted in this expression. We assume that the eigenvalues that correspond to different terms in this sum are distinct complex numbers defined up to a certain transformation (depending on which group is acting). The proportionality factor between matrix pencil and its Kronecker canonical form is also omitted. For instance, D_{2}(μ_{1}) ⊕ D_{1}(μ_{1}) ⊕ D_{1}(μ_{2}) corresponds to J_{3}(2) ⊕ J_{1}. The matrix pencil P ⊕⋯⊕P is shortly denoted by nP, where n is the number of terms in the sum. The matrix pencils L_{0} and R_{0} are not shown in the canonical form.

6. Matrix pencil bundles

A set X = {X_{α}} of subsets of a topological space X is called the stratification, if ∪X_{α} = X, the closure of each X_{α} is a union of elements of X, and X_{α}∩X_{β} = ∅ for α ≠ β. The sets X_{α} are called strata. We say that X_{α} covers X_{β}, if \overline{X_{α}} ⊃ \overline{X_{β}} and there is no X_{γ} other than X_{α} and X_{β} such that \overline{X_{α}} ⊃ \overline{X_{γ}} ⊃ \overline{X_{β}}.

Now let X be an algebraic manifold with an action of some algebraic group. We are interested in stratifications that have invariant strata. Certainly, such strata are the unions of orbits, and the orbital decomposition, that is, the set of all orbits is the finest possible invariant stratification. However, it typically has infinite number of strata. In the next few paragraphs we introduce the notion of matrix pencil bundle, which gives a natural stratification that has finite number of strata.

The term “bundle” originates in works of Arnold and deals with linear operatiors [1]. A bundle of linear operators is a set of linear operators that have fixed Jordan structure and varying eigenvalues. In other words, the linear operators ⊕J_{k_{i}}(α_{i}) and ⊕J_{k_{i}}(β_{i}) belong to the same bundle if and only if α_{i} = α_{j} implies β_{i} = β_{j} and vice versa for all i and j. Obviously, the number of bundles of linear operators is finite.

It is clear how to give similar definition for matrix pencils. Let P be a matrix pencil that is defined up to a transformation from GL_{n,m}, and let φ be a one-to-one mapping of \overline{ℂ} = ℂ∪\{∞\} into itself. The matrix pencil φ(P) is defined by

𝔏(φ(P)) = 𝔏(P),  (10)
ℜ(φ(P)) = ℜ(P),  (11)
𝔇(μ, φ(P)) = 𝔇(φ^{−1}(μ).P)  (12)

up to a a transformation from GL_{n,m}. The Kronecker canonical form of φ(P) is obtained from the Kronecker canonical form of P by replacement of the eigenvalues of P with their images under φ. Denote by Φ the set of one-to-one mappings of \overline{ℂ} into itself. The set

B(P) = ⋃_{φ∈Φ} φ(P)  (13)

is called a matrix pencils bundle. In what follows, the matrix pencils bundles are shortly referred to as bundles. Similarly to (13), we introduce the following notation for the unions of closures of orbits:

\overline{B}(P) = ⋃_{φ∈Φ} \overline{φ(P)}.  (14)

Obviously, B(P) ⊂ \overline{B}(P) ⊂ \overline{B(P)}.

Now let ψ be a mapping of \overline{ℂ} into itself (not necesserily one-to-one). Define ψ(P) by (10), (11) and

𝔇(μ, ψ(P)) = ⋃_{ν∈ψ^{−1}(μ)} 𝔇(ν, P).  (15)

The matrix pencil ψ(P) is defined up to a transformation from GL_{n,m} and is called the amalgam of P. In what follows we often use two special types of amalgams.

Let γ(z) be the mapping that takes all \overline{ℂ} to the point a, and let γ_{z_{0}}(z) be the mapping that takes \overline{ℂ}\\{z_{0}\} to the point a and z_{0} to some other point b for some z_{0} ∈ \overline{ℂ}. If P is a matrix pencil and μ is an eigenvalue of P then the amalgams γ(P) and γ_{μ}(P) are called the main and the submain amalgams, respectively.

Assume that P is a Kronecker canonical form, and let 𝔖(P) ⊂ \overline{ℂ} be the set of eigenvalues of P. Consider the set Z = {(x_{1}, …, x_{s}) ∈ ℂ^{s} | x_{i} ≠ x_{j} if i ≠ j}, where s = |𝔖(P)|, and the mapping f : Z × GL_{n,m} → ℂ^{n,m,2}, that takes each pair ((x_{1}, …, x_{s}), g) to a matrix pencil obtained from P by replacement of the eigenvalues of P with x_{1}, …, x_{s} and consequent application of the action of g ∈ GL_{n,m}. This mapping is polynomial and has constant rank. Since the dimension of stabilizer of a matrix pencil depends on the partitions 𝔇(μ, P) but not on the eigenvalues themselves, the image of f, that is, B(P) is an irreducible regular algebraic manifold, whose codimension is given by

cod(ℂ^{n,m,2}, B(P)) = cod(ℂ^{n,m,2}, O(P)) − |𝔖(P)|.  (16)

The codimension of orbit, which appears on the right side of (16), can be calculated as follows [4]. Let P and Q be n_{1} × m_{1} and n_{2} × m_{2} matrix pencils. Define

⟨P, Q⟩ = dim{(A, B) ∈ (ℂ^{n_{1}} ⊗ ℂ^{n_{2}}) × (ℂ^{m_{1}} ⊗ ℂ^{m_{2}}) | PA = BQ}.

Then the dimension of stabilizer of n × m matrix pencil P in GL_{n}(ℂ) × GL_{m}(ℂ) is equal to ⟨P, P⟩, and the codimension of its orbit in ℂ^{n,m,2} is equal to ⟨P, P⟩ − (n − m)^{2}. One can prove that ⟨P, Q⟩ = ∑ ⟨P_{i}, Q_{j}⟩, if P = ⊕P_{i} and Q = ⊕Q_{j}. For μ_{1} ≠ μ_{2} we have ⟨D_{k}(μ_{1}), D_{j}(μ_{2})⟩ = 0; for the other pairs of indecomposable matrix pencils the values of ⟨P, Q⟩ are given in the following table.

\begin{aligned}& L_{k} & R_{k} & D_{k}(μ) \\ L_{j} & (j − k + 1)_{+} & j + k & j \\ R_{j} & 0 & (k − j + 1)_{+} & 0 \\ D_{j}(μ) & 0 & k & min(j, k) \end{aligned}  (17)

This gives an efficient method for computation of orbit’s codimension from the Kronecker canonical form. By dropping L_{0} and R_{0} in the Kronecker canonical form we implicitly use the natural embedding of ℂ^{n,m,2} into ℂ^{N,M,2} for N ≥ n and M ≥ m. Codimensions of orbits, however, depend on the dimension of the enveloping space. This motivates the following definition. A matrix pencil P is said to be a veritable n × m matrix pencil, if l_{0}(P) = r_{0}(P) = 0. Veritability of a matrix pencil depends on the dimension of the representation. The n × m matrix pencils that are not veritable n × m matrix pencils are veritable matrix pencils of smaller orders.

It is convenient to reduce the calculation of the orbit’s codimension to the case of veritable matrix pencils. Let P be a veritable n×m matrix pencil , N ≥ n, M ≥ m, D = N − M, d = n − m. Simple algebra proves that the codimension of GL_{N,M} -orbit of P is given by

cod(ℂ^{N,M,2}, GL_{N,M}P) − cod(ℂ^{n,m,2}, GL_{n,m}P) = d^{2} − D^{2} + N(N − n) + M(M − m).

7. Degenerations of matrix pencil bundles

For any partition n, put J(μ, n) = ⊕J_{n_{i}}(μ), μ ∈ ℂ and J (μ, n) = ⊕D_{n_{i}}(μ), μ ∈ \overline{ℂ}; J(μ, n) is a matrix, J (μ, n) is a matrix pencil.

Lemma 7.1. Let A be an upper triangular block matrix with blocks A_{1}, A_{2}, …, A_{m} on the diagonal. If A_{i} is conjugate to J(μ, n_{i}) for all i then A belongs to the closure of SL_{n}(ℂ)-orbit of J(μ, ∑ n_{i}). If A_{i} is conjugate to J(μ_{i}, n_{i}) for all i and μ_{i} ≠ μ_{j} for i ≠ j then A is conjugate to ⊕J(μ_{i}, n_{i}).

Proof. The first statement is found in [9]. The proof of the second statement is trivial.

Corollary 7.2. Let P be an upper triangular block matrix pencil with blocks P_{i} on the diagonal. If P_{i} is equivalent to J (μ, n_{i}) for all i then P belongs to the closure of GL_{n,m}-orbit of J (μ, ∑ n_{i}). If P_{i} is equivalent to J (μ_{i}, n_{i}) for all i and μ_{i} ≠ μ_{j} for i ≠ j then P and ⊕J (μ_{i}, n_{i}) are equivalent.

Lemma 7.3. Let γ_{1}(ε), …, γ_{s}(ε) be a set of complex numbers that are distinct for all ε ∈ (0, ε_{0}] and γ_{s}(ε) → 0 when ε → 0 for all i. There exist matrices C_{ε} such that C_{ε}(J_{k_{1}}(γ_{1}(ε)) ⊕ ⋯ ⊕ J_{k_{s}}(γ_{s}(ε)))C_{ε}^{−1} → J_{k_{1}+···+k_{s}}(0) when ε → 0.

Proof. Find a_{1}(ε), …, a_{n}(ε) from the equiation λ^{n} − a_{1}(ε)λ^{n−1} − a_{2}(ε)λ^{n−2} − ⋯ − a_{n}(ε) = (λ − γ_{1}(ε))^{k_{1}} … (λ − γ_{s}(ε))^{k_{s}} and define

B_{ε} = (\begin{matrix}a_{1}(ε) & 1 \\ a_{2}(ε) & 0 & … \\ … & & … & 1 \\ a_{n}(ε) & & & 0 \end{matrix}) .

Clearly, det(λE − B_{ε}) = (λ − γ_{1}(ε))^{k_{1}} … (λ − γ_{s}(ε))^{k_{s}} and the rank of λE − B_{ε} is greater then or equal to n − 1 for all λ ∈ ℂ. Therefore, the Jordan normal form of B_{ε} consists of blocks J_{k_{1}}(γ_{1}(ε)), …, J_{k_{s}}(γ_{s}(ε)), i.e. B_{ε} = C_{ε}^{−1}J_{k_{1}}(γ_{1}(ε)) ⊕ ⋯ ⊕ J_{k_{s}}(γ_{s}(ε))C_{ε} for some C_{ε} ∈ SL_{n}(ℂ). To complete the proof, it remains to note that all a_{i}(ε) tend to 0 when ε → 0.

Lemma 7.4. If P is an upper triangular block matrix pencil with blocks P^{l}, P^{reg}, P^{r} in the diagonal (in this order!), P^{reg} is a regular matrix pencil, P^{l} contains only the blocks L_{i}, and P^{r} contains only the blocks R_{j} then P is equivalent to P^{l} ⊕ P^{reg} ⊕ P^{r}.

Proof. For simplicity, assume that P is

(\begin{matrix}P^{l} & A \\ 0 & P^{reg} \end{matrix}) .

To prove the equivalence of P and P^{l} ⊕ P^{reg}, it is enough to find matrices X and Y such that P^{l}X + Y P^{reg} + A = 0. We prove this statement for P^{l} = L_{i} and P^{reg} = D_{j}(μ). Consider the linear operator that takes each pair of matrices (X, Y) to the matrix pencil L_{i}X + Y D_{j}(μ). This operator acts on (2ij + j)-dimensional complex vector space. According to (17), the dimension of its kernel is j. Therefore the dimension of its image is 2ij, and, therefore, the desired matrices X and Y exist for any i × j matrix pencil A.

Theorem 7.5. The closure of B(P) is the union of closures of its orbits and closures of their amalgams. That is,

\overline{B(P)} = ⋃_{ψ} \overline{B}(ψ(P)),

where ψ runs over the set of all mappings of \overline{ℂ} into itself.

Proof. By lemma 7.3, B(ψ(P)) ⊂ \overline{B(P)}. Therefore, \overline{B}(ψ(P)) ⊂ \overline{B(P)} for any amalgam ψ. Now we need to prove that if P_{n} ∈ B(P) is a sequence of matrix pencils that converges to P_{∗} then P_{∗} belongs to \overline{B}(ψ(P)) for some ψ. Assume that P_{1} = P and denote the eigenvalues of P_{n} by μ_{n1}, …, μ_{ns}. Since \overline{ℂ} is compact, we may assume that μ_{ni} converges to some μ_{i} for all i = 1 … s. Consider the mapping ψ (defined on the set of eigenvalues of P_{1}) that takes each μ_{1i} to μ_{i}. It follows from Ivasawa decomposition in the groups GL_{n}(ℂ) and GL_{m}(ℂ) that there exist unitary matrices P_{n} and Q_{n} such that

P_{n}P_{n}Q_{n}^{−1} = (\begin{matrix}P_{n}^{l} & ∗ & ∗ \\ 0 & P_{n}^{reg} & ∗ \\ 0 & 0 & P_{n}^{r} \end{matrix}) ,  (21)

where P_{n}^{reg} is an upper triangular matrix with numbers a_{in} + λb_{in} in the diagonal, b_{in}/a_{in} = μ_{in}, where μ_{in} that are equal to each other run in succesion, P^{l} contains only the blocks L_{i}, P^{r} contains only the blocks R_{j}, and P_{n}^{l} ⊕ P_{n}^{reg} ⊕ P_{n}^{r} is equivalent to P_{n}. Decomposition (21) is called the generalized Shur form of a matrix pencil [24].

Since P_{n} and Q_{n} are unitary, we may assume that their limits exist. The numbers a_{in} and b_{in} are uniformely bounded by i, as P_{n} is converging. Therefore, we may assume that a_{in} → a_{i} and b_{in} → b_{i}. Put α_{in} = a_{in} − a_{i}, β_{in} = b_{in} − b_{i} and consider the matrix pencil

P′_{n} = (\begin{matrix}P_{n}^{l} & ∗ & ∗ \\ 0 & P_{n}^{reg} − Q_{n} & ∗ \\ 0 & 0 & P_{n}^{r} \end{matrix}) ,

where Q_{n} is a diagonal matrix pencil with α_{in}+λβ_{in} on the diagonal. By lemma 7.4 P′_{n} is equivalent to P_{n}^{l}⊕(P_{n}^{reg}−Q_{n})⊕P_{n}^{r}. By corollary 7.2, we have P′_{n} ∈ \overline{O(ψ(P))} for all n and, since the limit of Q_{n} is 0, (lim P_{n})P_{∗}(lim Q_{n})^{−1} ∈ \overline{O(ψ(P))}. Therefore, P_{∗} ∈ \overline{B}(ψ(P)). The proof is now completed.

Remark 7.6. The sequences a_{in} and b_{in} are uniformely bounded by i since P_{n} converges. Therefore, a_{i} and b_{i} are both finite. What if a_{i} and b_{i} are both equal to zero? Then P_{n}^{reg} − Q_{n} would not be a regular matrix pencil and the premise of lemma 7.4 would not hold. In this case we multiply P_{n} by a diagonal matrix T_{n} = diag(t_{in}) to prevent a_{in} and b_{in} from tending to zero simultaneously. Then, P_{n}P_{n}Q_{n}^{−1} − T_{n}^{−1}Q_{n} ∈ \overline{B}(ψ(P)), but T_{n}^{−1}Q_{n} → 0 since t_{in} must tend to infinity.

8. Degenerations of orbits of GL_{n,m} and GL_{n,m,2}

Description of closures of matrix pencil bundles is now reduced to description of closures of GL_{n,m}-orbits. The closures of GL_{n,m}-orbits have been described in [21]. In particular, the matrix pencil Q belongs to the closure of GL_{n,m}-orbit of the matrix pencil P if and only if

ℜ(P) + nrk(P) ≥ ℜ(Q) + nrk(Q),
𝔏(P) + nrk(P) ≥ 𝔏(Q) + nrk(Q),
𝔇(μ, P) + r_{0}(P) ≤ 𝔇(μ, Q) + r_{0}(Q)

hold for all μ ∈ \overline{ℂ} [21]. The Kronecker canonical form of Q can be obtained from the Kronecker canonical fom of P by the following transformations [21]:

Ia. R_{j} ⊕ R_{k} → R_{j−1} ⊕ R_{k+1}, 1 ≤ j ≤ k,

Ib. L_{j} ⊕ L_{k} → L_{j−1} ⊕ L_{k+1}, 1 ≤ j ≤ k,

IIa. R_{j+1} ⊕ D_{k}(μ) → R_{j} ⊕ D_{k+1}(μ), j, k ≥ 0, μ ∈ \overline{ℂ},

IIb. L_{j+1} ⊕ D_{k}(μ) → L_{j} ⊕ D_{k+1}(μ), j, k ≥ 0, μ ∈ \overline{ℂ},

III. D_{j−1}(μ) ⊕ D_{k+1}(μ) → D_{j}(μ) ⊕ D_{k}(μ), 1 ≤ j ≤ k, μ ∈ \overline{ℂ},

IV. ⨁_{i=1}^{s} D_{k_{i}}(μ_{i}) → L_{p} ⊕ R_{q}, μ_{i} ≠ μ_{j} for i ≠ j, μ_{i} ∈ \overline{ℂ}, p + q + 1 = ∑_{i=1}^{s} k_{i}.

The transformations I–IV are not minimal, but the list of minimal transformations is easily derived from them [6].

In section 2. we have shown that GL_{n,m,2}-orbit of a matrix pencil P is a union of GL_{n,m}-orbits φ(P) over all linear fractional transformations of \overline{ℂ}. The following lemma describes the limits of sequences of linear fractional transformations.

Lemma 8.1. For any sequence of linear fractional transformations of \overline{ℂ} there exist a subsequence that converges in pointwise topology. The limit of a sequence of linear fractional transformations is either a linear fractional transformation or a mapping that is constant on \overline{ℂ} except for, may be, one point.

Proof. Let φ_{R_{k}}(z) = (r_{21}^{k} + r_{22}^{k}z)/(r_{11}^{k} + r_{12}^{k}z) be a sequence of linear fractional transformations. Since r_{ij}^{k} are defined up to proportionality, we may assume that r_{ij}^{k} converges to some r_{ij} for all i and j and r_{ij} are not all equal to zero. Passing to φ_{R_{k}}(1/z) or (φ_{R_{k}}(z))^{−1}, we may assume that r_{12} ≠ 0. If matrix (r_{ij}) is degenerate, then the limits of φ_{R_{k}}(z) are equal to each other for all z ∈ \overline{ℂ} except for, may be, z = −r_{11}/r_{12}. If matrix (r_{ij}) is non-degenerate, then the limit is a linear fractional transformation.

Now let us go back to the definition of matrix pencil bundles. If in (13) we allowed φ to be only a linear fractional transformation then the bundles would be exactly GL_{n,m,2}-orbits. This observation allows us to use the proof of theorem 7.5 for the following theorem.

Theorem 8.2.

\overline{B(P)} = \overline{B}(P) ∪ ( ⋃_{μ∈𝔖(P)} \overline{B}(γ_{μ}(P))) ∪ \overline{B}(γ(P)),

where B(P) = GL_{n,m,2}P and \overline{B}(P) = GL_{n,m,2}(\overline{GL_{n,m}P}).

Proof. Let μ_{1}, …, μ_{s} be the set of eigenvalues of P. It follows from lemma 7.3 that B(γ(P)) ⊂ \overline{B(P)} since μ_{1}, …, μ_{s} are mapped to εμ_{1}, …, εμ_{s} by the linear fractional transformation z ↦ εz. Similarly, B(γ_{μ}(P)) ⊂ \overline{B(P)}. The proof of the converse is similar to one of theorem 7.5; the difference is that if two eigenvalues coalesce then all eigenvalues do except for, may be, one (lemma 8.1).

Thus, the regular parts of GL_{n,m,2}-orbits either coalesce all together, or coalese all but one, or do not coalese at all.

9. Degenerations of orbits of SL_{n,m} and SL_{n,m,2}.

We have shown that SL_{n,m}-orbits of perfect matrix pencils are closed, while SL_{n,m}-orbits of imperfect singular matrix pencils coincide with their GL_{n,m}-orbits (theorem 3.3). We now have to describe degenerations of SL_{n,m}-orbits of regular matrix pencils. It is enough to determine which of the transformations I–IV (section 8.) can be executed under the action of SL_{n,m}. First of all, transformations I–II are not applicable to regular matrix pencils. The transformation IV is also not applicable, since the coefficients of the binary form θ(P) are invariant and therefore the closure of SL_{n,m}-orbit of a regular matrix pencil consists of regular matrix pencils only. Transformation III, indeed, is applicable to regular matrix pencils; it corresponds to regrouping of Jordan blocks under the adjoint representation of SL_{n}(ℂ).

Now we focus on SL_{n,m,2}-orbits. As before, we may confine ourselves to SL_{n,m,2}-orbits of regular matrix pencils because SL_{n,m}-orbits and GL_{n,m}-orbits of imperfect singular matrix pencils coincide (theorem 3.4). Recall that SL_{n,m,2}-orbit of a regular matrix pencil is defined uniquely by the partitions 𝔇(μ_{j}, P) and the set of eigenvalues, which are defined up to a linear fractional transformation that satisfies (5). The condition (5) is equivalent to

∏_{i=1}^{n}(r_{11} + r_{12}μ_{i}) = (det R)^{n/2}  (23)

for a regular matrix pencil P = E + λD, where D is an upper triangular matrix with μ_{1}, …, μ_{n} on the diagonal. Let φ_{R}(z) be a linear fractional transformation that satisfies (23). Since

φ_{R}(z) − φ_{R}(w) = \frac{det R (z − w)}{(r_{11} + r_{12}z)(r_{11} + r_{12}w)}  (24)

holds for any z and w, we have

W (φ_{R}(μ_{1}), …, φ_{R}(μ_{n})) = \frac{(det R)^{\frac{n(n−1)}{2}} W (μ_{1}, …, μ_{n})}{(∏(r_{11} + r_{12}μ_{i}))^{n−1}},  (25)

where W (z_{1}, …, z_{n}) = ∏_{i>j}(z_{i} − z_{j}) is the Wandermond determinant. It follows

from (23) that

W (φ_{R}(μ_{1}), …, φ_{R}(μ_{n})) = W (μ_{1}, …, μ_{n}).  (26)

Theorem 9.1. SL_{n,m,2}-orbit of a regular n × n matrix pencil that has exactly n distinct eigenvalues is closed.

Proof. Let g_{k} = (P_{k}, Q_{k}, R_{k}) be a sequence of elements of SL_{n,m,2} such that lim g_{k}P = P^{∗}. Denote the eigenvalues of P by μ_{1}^{∗}, …, μ_{n}^{∗}, and let μ_{i}^{∗} be the limits of φ_{R_{k}}(μ_{i}) when k → ∞. We may assume that μ_{i} and μ_{i}^{∗} are finite for all i = 1 … n. It follows that W (μ_{1}^{∗}, …, μ_{n}^{∗}) = W (μ_{1}, …, μ_{n}) and, therefore, μ_{1}^{∗}, …, μ_{n}^{∗} are distinct and the limit of φ_{R_{k}} is a linear fractional transformation. Since SL_{n,m}-orbit of P is closed, it follows that P and P^{∗} are SL_{n,m,2}-equivalent.

This theorem has a simplier proof. The orbit of a regular matrix pencil whose eigenvalues are distinct is a pre-image of the orbit of a binary form that doesn’t have multiple factors. Such orbits are known to be closed. However, this proof is applicable only when the eigenvalues are distinct, but there are other closed orbits. In the proof of theorem 9.1 we essentially used the fact that W (μ_{1}, …, μ_{n}) is non-zero. Now we introduce another approach that works for matrix pencils with multiple eigenvalues.

Lemma 9.2. Let n = (n_{1}, …, n_{s}) be a partition of an even integer n. The integer n_{1} is smaller than or equal to the half of n if and only if there exist a non-oriented graph whose vertices are n_{1}, …, n_{s} and the number of edges coming from n_{i} is equal to n_{i} for all i.

Proof. Let us translate this statement to the language of Ferrer diagrams. The Ferrer diagram of n can be built step by step from an empty diagram by picking any two columns and adding one new cell to each of them. Equivalently, the Ferrer diagram of n can be sequentially disassembled by picking any two columns on each step and deleting one cell in each of them. Clearly, if each column gets one cell at one time then none of the columns gets more than a half of the total number of cells. This proves the necessary part. The proof of the converse is by induction over n. Consider three cases:

1. n_{1} > n_{2}. Delete the upper cells in the first column and (some) other column without loss of monotonicity of n. The partition obtained has the same property: the number of cells in its first column is smaller than or equal to the total number of cells.

2. n_{1} = n_{2} > n_{3}. Delete the upper cells in the first two columns. Then the number of cells in the first column is again smaller than or equal to the total number of cells.

3. n_{1} = n_{2} = n_{3} = n_{s} > n_{s+1}. In this case n_{1} ≤ n/3. Therefore, n_{1} < n/2 − 1 for n > 6. Now delete the upper cells in the (s−1)-th and the s-th columns. The number of cells in the first column of the partition obtained is smaller than or equal to the total number of cells. The proof is trivial for n ≤ 6.

[figure: a; b; c; 1; 1; 1; 3; 2; 2; 2; 2; 2; 2; 2; 2]

Figure 1: The graphs corresponding to the partitions (3, 1, 1, 1) and (2, 2, 2, 2).

The non-oriented graph whose existence is guaranteed by previous lemma is not always well-defined. Figure 1 illustrates this for the partitions (3, 1, 1, 1) (a) and (2, 2, 2, 2) (b and c). A non-oriented graph is said to be a star, if all its edges share a common node. Obviously, the graph built in lemma 9.2 is a star if and only if n_{1} = n/2 and in this case it is defined unambiguously.

In what follows, only the main and the submain amalgams of SL_{n,m,2}-orbits are considered. The number of distinct eigenvalues of these amalgams is smaller than or equal to two. Therefore, SL_{n,m,2}-orbits of γ_{μ}(P) and γ(P) are welldefined, since every pair of complex numbers can be taken to another pair by a linear fractional transformation that satisfies (5).

We have shown that the regular matrix pencil P is nilpotent if and only if the binary form θ(P) is nilpotent. It is known that the binary form is nilpotent if and only if it has a linear factor whose multiplicity is greater than the half of the degree of the form. Hence, the regular matrix pencil P is nilpotent if and only if it has an eigenvalue whose multiplicity is greater than the half of the order of the matrix pencil. The following theorem describes closures of non-nilpotent SL_{n,m,2}-orbits.

Theorem 9.3. Put B(P) = SL_{n,m,2}P and \overline{B}(P) = SL_{n,m,2}(\overline{SL_{n,m}P}). If the multiplicities of all eigenvalues of a regular matrix pencil P are strictly smaller than the half of its order then \overline{B(P)} = \overline{B}(P). If the multiplicitiy of the eigenvalue μ is equal to the half of the order of P then \overline{B(P)} = \overline{B}(P) ∪ \overline{B}(γ_{μ}(P)).

Proof. Recall the proof of theorem 9.1. Consider the sequence P_{k} of elements of SL_{n,m,2}-orbit of P that converges to P^{∗} and denote by μ_{i}^{∗} the limit of φ_{R_{k}}(μ_{i}). Since P is not nilpotent, the sequence of binary forms θ(P_{k}) converges to a nonzero limit. Therefore, μ_{i}^{∗} are the eigenvalues of P^{∗}. Assume that μ_{1}, …, μ_{s} are distinct eigenvalues of P. Their multiplicities form a partition, which we denote by n = (n_{1}, n_{2}, …).

(i) Case n_{1} < n/2. If n is an odd number then multiply n by 2. Denote by E the set of edges of the graph Γ that was constructed in lemma 9.2. We now follow the proof of theorem 9.1 with one exception: instead of W we consider

Ŵ(z_{1}, …, z_{s}) = ∏_{[i,j]∈E}(z_{i} − z_{j}),

where [i, j] denotes the edge connecting the i-th and the j-th nodes, i < j. It follows from (24) that

Ŵ(φ_{R}(μ_{1}), …, φ_{R}(μ_{s})) = \frac{(det R)^{n}Ŵ(μ_{1}, …, μ_{s})}{(∏(r_{11} + r_{12}μ_{i})^{n_{i}})^{2}}.

For any fractioal transformation φ_{R}(z) that satisfies (23) we have Ŵ(φ_{R}(μ_{1}), …, φ_{R}(μ_{s})) = Ŵ(μ_{1}, …, μ_{s}). Therefore, Ŵ(μ_{1}^{∗}, …, μ_{s}^{∗}) = Ŵ(μ_{1}, …, μ_{s}). Suppose that the limit of φ_{R_{k}} is not a linear fractional transformation. Then all μ_{i}^{∗} are equal to each other but may be one. Note that Γ is not a star when n_{1} < n/2. If we delete one node along with all its edges then there will remain at least one edge. Since this edge corresponds to a factor in Ŵ(μ_{1}^{∗}, …, μ_{s}^{∗}), we have Ŵ(μ_{1}^{∗}, …, μ_{s}^{∗}) = 0, and therefore Ŵ(μ_{1}, …, μ_{s}) = 0. This contradicts to the assumption that μ_{1}, …, μ_{s} are distinct. Thus, the limit of φ_{R_{k}} is a linear fractional transformation, and P^{∗} is SL_{n,m,2}-equivalent to a matrix pencil from \overline{SL_{n,m}P}.

(ii) Case n_{1} = n/2. Using the same arguments one can prove that the eigenvalue μ whose multiplicity is n/2 cannot coalesce with the others. It follows that \overline{B(P)} ⊂ \overline{B}(P) ∪ \overline{B}(γ_{μ}(P)). It now remains to prove that, indeed, \overline{B}(γ_{μ}(P)) ⊂ \overline{B(P)}. It is convenient to change the notation now. Let μ be equal to 0, and denote the other eigenvalues by μ_{1}, …, μ_{s}. Consider the linear fractional transformation

φ_{k}(z) = \frac{b_{k}c_{k}z}{1 + c_{k}z} .  (27)

Then (23) is equivalent to

b_{k}^{n/2} = \frac{(1 + c_{k}μ_{1}) … (1 + c_{k}μ_{s})}{c_{k}^{n/2}}.  (28)

The degree of the denominator is equal to the degree of the numerator (s = n/2). If c_{k} → ∞ when k → ∞, then b_{k} → (μ_{1} … μ_{s})^{2/n} and φ_{k}(z) → (μ_{1} … μ_{s})^{2/n} ≠ 0 for any z ∈ ℂ\{0}, and φ(0) = 0. The proof is now completed.

Theorem 9.4. Let B(P) and \overline{B}(P) be as in theorem 9.3, and let R and S denote the sets of regular and singular n × n matrix pencils, respectively, and consider a nilpotent regular matrix pencil P. Then \overline{B(P)} ∩ R = \overline{B}(P) ∪ \overline{B}(γ(P)) and \overline{B(P)} ∩ S = \overline{GL_{n,m}(γ(P))} ∩ S.

Proof. First we prove that if some of the eigenvalues coalesce then they all do. Suppose that the eigenvalue that has the greatest multiplicity is equal to zero, and denote the other eigenvalues by μ_{1}, …, μ_{s}. We may assume that φ_{R_{k}}(0) = 0 for all k. Then φ_{R_{k}} has form (27). The limit of φ_{R_{k}} is not a linear fractional transformation if and only if the limit of det(R_{k}) is equal to zero or is infinite. If it is equal to zero then all μ_{i}^{∗} are equal to zero. If it is infinite then it follows from (28) that c_{k} → ∞ and b_{k} → 0, since the degree of the nominator is smaller than the degree of the denominator (s < n/2). Thus, φ_{k}(z) → 0 for all z ∈ ℂ.

If the limit of φ_{R_{k}} is a regular matrix pencil that doesn’t belong to \overline{B}(P) then its eigenvalues are all equal to each other. That is, \overline{B(P)} ∩ R ⊂ \overline{B}(γ(P)). It is clear that \overline{B}(γ(P)) ⊂ \overline{B(P)} ∩ R. If the limit of φ_{R_{k}} is a singular matrix pencil then there exist unitary transformations P_{n} and Q_{n} that take P_{n} to the generalized Shur form. As in theorem 7.5 we can substract a matrix pencil Q_{n} from P_{n}P_{n}Q_{n}^{−1} to make the difference belong to \overline{GL_{n,m}(γ(P))}. Now it remains to prove that \overline{GL_{n,m}(γ(P))}∩S ⊂ \overline{B(P)}. It is enough to prove that \overline{GL_{n,m}(γ(P))}∩S ⊂ \overline{B(γ(P))}. The Kronecker canonical form of a singular matrix pencil that is covered by GL_{n,m}-orbit of a regular matrix pencil can be obtained using transformation IV (section 8.). The matrix pencils

(\begin{matrix}1 & λ \\ & ... & ... \\ & & 1 & λ \\ & & & ... & ... \\ & & & & 1 & λ \end{matrix}) and (\begin{matrix}1 & λ \\ & ... & ... \\ & & ε & λ \\ & & & ... & ... \\ & & & & 1 & λ \end{matrix})

are SL_{n,m,2}-equivalent for all ε ≠ 0 (the transformation consists in multiplication of the first matrix by diag(ε, ε, …, ε^{1−n}, …, ε) and consequent application of the action of diag(ε^{−1}, ε)). Thus, the degeneration D_{n}(0) → L_{p}⊕R_{q} can be performed by the action of SL_{n,m,2}.

10. Minimal degenerations

So far we have obtained the criteria for a matrix pencil to belong to the closure of an orbit (or bundle) of another matrix pencil. In order to describe the hierarchy of closures, we need to come up with a description of the covering relationship for all cases discussed in sections 7.–9.. For GL_{n,m}-orbits and matrix pencil bundles it has been done in [6].

The orbit of P covers the orbit of Q if and only if the partitions 𝔇(μ, Q), ℜ(Q), and 𝔏(Q) are obtained from 𝔇(μ, P), ℜ(P), and 𝔏(P), respectively, by one of the following transformations:

A1. Minimal lowering of 𝔏(P) or ℜ(P) that doesn’t affect the leftmost column.

A2. Deletion of the rightmost standalone cell of ℜ(P) or 𝔏(P) and simultaneous addition of a new cell to 𝔇(μ, P) for some μ ∈ \overline{ℂ}.

A3. Minimal hightening of 𝔇(P).

A4. Deletion of all cells in the lowest raw of each 𝔇(μ, P) and simultaneous addition of one new cell to each 𝔩_{p}(P), p = 0, …, t and 𝔯_{q}(P), q = 0, …, k− t−1, where k is the number of deleted cells, such that every non-zero column is given at least one cell.

The rule A1 corresponds to the transformations Ia and Ib, the rule A2 corresponds to the transformations IIa and IIb, the rule A3 corresponds to the transformation III and the rule A4 corresponds to the transformation IV. A good set of figures illustrating the respective transformations of the Ferrer diagrams is found in [6].

The bundle of P covers the bundle of Q if and only if the partitions 𝔇(μ, Q), ℜ(Q), and 𝔏(Q) are obtained from 𝔇(μ, P), ℜ(P), and 𝔏(P), respectively, by the rules A1, B2, A3, B4 and B5:

B2. Same as rule A2, but start with a new set of cells for a new eigenvalue (otherwise it could have been done using A2 and then B5).

B4. Same as rule A4, but appply only if there is just one eigenvalue or if all eigenvalues have at least 2 Jordan blocks (otherwise it could have been done using B5 and then A4).

B5. Union of any two 𝔇(μ, P).

Here we list the similar set of rules for minimal degenerations of SL_{n,m,2} and GL_{n,m,2}-orbits. No proof is necessary as they are obtained from ones above in a very straightforward way.

The GL_{n,m,2}-orbit of P covers the GL_{n,m,2}-orbit of Q if and only if 𝔇(μ, Q), ℜ(Q), and 𝔏(Q) are obtained from 𝔇(μ, P), ℜ(P), and 𝔏(P), respectively, by the rules A1, C2, A3, C4, and C5:

C2. Same as rule A2, but start a new set of cells for a new eigenvalue if the matrix pencil has less than three eigenvalues.

C4. Same as rule A4, but apply only if there is just one eigenvalue or if all eigenvalues have at least 2 Jordan blocks.

C5. Union of all 𝔇(μ, P) but one.

The SL_{n,m,2}-orbit of P covers the SL_{n,m,2}-orbit of Q if and only if 𝔇(μ, Q), ℜ(Q), and 𝔏(Q) are obtained from 𝔇(μ, P), ℜ(P), and 𝔏(P), respectively, by the following rules:

1. rules A1, C2, A3, C4 and C5 , if the matrix pencils is imperfect and singular;

2. rules A3, D4 and D5

D4. Rule A4 and consequent union of all 𝔇(μ, P).

D5. Union of all 𝔇(μ, P),

if the matrix pencils is nilpotent and regular;

3. rules A3 and D6

D6. Union of all 𝔇(μ, P) except for ones, whose eigenvalue’s multiplicity is n/2.

if the matrix pencil is regular and has an eigenvalue with mutiplicity n/2;

4. rule A3 for all other regular matrix pencils.

11. Null cone

Recall that the null cone is the union of all nilpotent orbits. The main focus of this paragraph is the null cone of SL_{n,m,2}.

Theorem 11.1. The null cone of SL_{n,m,2} is irreducible and contains an open orbit for n ≠ m.

Proof. Suppose m > n and put d = m − n. Condiser the matrix pencil P = dL_{k} ⊕ (n mod d)L_{k+1}, where k = [n/d]. If d divides n then P is a perfect matrix pencil and the null cone is the border of its GL_{n,m,2}-orbit. The Ferrer diagram of 𝔏(P) is a rectangle with k columns and d raws. If d > 1 then only the rule A1 is applicable to this diagram. If d = 1 then the only applicable rule is C2. The corresponding degenerations are (d − 2)L_{k} ⊕ L_{k−1} ⊕ L_{k+1} and (d − 1)L_{k−1} ⊕ J_{1}. Therefore, the orbit of P covers only one SL_{n,m,2}-orbit, since the other GL_{n,m,2}-orbits and SL_{n,m,2}-orbits coincide (theorem 3.4). Then the null cone is irreducible and has an open orbit. If d doesn’t divide n then GL_{n,m,2}-orbits and SL_{n,m,2}-orbits coincide and the null cone is ℂ^{n,m,2}. The reader will easily prove that the closure of the orbit of P contains all other orbits.

Theorem 11.2. The null cone of SL_{n,n,2} is irreducible.

Proof. A regular matrix pencil is nilpotent if and only if multiplicity of one of its eigenvalues is greater than the half of its order. Then the null cone of SL_{n,n,2} is the union of the set of the matrix pencil bundles that have an eigenvalue with multiplicity n/2 and the set of singular matrix pencils. There is only one bundle (namely, J_{[n/2]+1} ⊕ (n − [n/2] − 1)J_{1}) whose closure is the entire null cone. It now remains to note that the bundles are irreducible.

12. Hierarchy of closures

It turns out that the hierarchy of closures of matrix pencil bundles has an interesting property: the matrix pencil bundles can be subdivided into three classes — left, central and right, which are defined by 𝔏(P) > ℜ(P), 𝔏(P) = ℜ(P), and 𝔏(P) < ℜ(P), respectively, such that if P_{1} is left, P_{2} is right, and \overline{B(P_{1})} ⊃ \overline{B(P_{2})} then there is a central matrix pencil Q such that \overline{B(P_{1})} ⊃ \overline{B(Q)} ⊃ \overline{B(P_{2})}. In other words, matrix pencil bundles from the left and the right classes do not cover each other. Also, B(Q) covers B(P) if and only if B(Q^{⊤}) covers B(P^{⊤}). Thus, we can save some space in figures illustrating the closures’ hierarchy by showing only left and central matrix pencils. Sadly, this symmetry breaks for 6 × 7 matrix pencils (see #223 in table 1). Moreover, in higher dimensions there exist matrix pencils such that 𝔏(P) and ℜ(P) are not comparable at all (for example, L_{0} ⊕L_{3} ⊕2R_{1}).

The classification of bundles and GL_{n,m,2}- and SL_{n,m,2}-orbits of matrix pencils is summarized in table 1 with the following conventions. The consecutive indexing for matrix pencils of all sizes is used. Only the left and the central veritable matrix pencils (except for #223) are shown. The right veritable matrix pencils are denoted by k^{⊤}, where k is the index of the corresponding transposed left matrix pencil. For example, 7^{⊤} stands for R_{2}. The columns “n” and “m” have obvious meaning. The column “c” contains codimensions of matrix pencil bundles in ℂ^{n,m,2} (that is in the space where they are veritable). The column “bundle” lists indices of matrix pencil bundles that are covered by the given matrix pencil bundle; k^{∗} means that the given bundle covers both k and k^{⊤}. For instance, R_{1} ⊕ L_{1} ⊕ J_{1} (# 25) covers L_{1} ⊕ 2J_{1} and R_{1} ⊕ 2J_{1}. The column “GL_{n,m,2}-orbit” lists the indices of GL_{n,m,2}-orbits that are covered by the given GL_{n,m,2}-orbit. If a cell of this colums is empty then its content was the same as in “bundle”. The last column lists the indices of SL_{n,m,2}-orbits that are covered by SL_{n,m,2}-orbit of P, where P is a (veritable) n × m matrix pencil. The indices of SL_{n,m,2}-orbits that are covered by the orbit of of P under the action of SL_{n,m,2} for larger n and m are found in the column “GL_{n,m,2}-orbit”. For instance, the SL_{n,m,2}-orbit of the 6×7 matrix pencil #228 covers the SL_{n,m,2}-orbit of the 6×6 matrix pencil #168. The symbol of empty set stands for the closed orbits.

The hierarchy of closures of matrix pencil bundles is shown in figures 2–5. The dots correspond to the matrix pencil bundles; if the canonical form is not shown in the figure then it is found in table 1. In all figures except for figure 2 (a) only the left and the central matrix pencils are shown. Two nodes are connected with an edge if and only if the matrix pencil bundle that corresponds to the node drawn above covers the matrix pencil bundle that corresponds to the node drawn below. The matrix pencil bundles, whose corresponding nodes are drawn on the same level, have equal codimensions, which are indicated by framed numbers on the side of the diagram.

[figure: J_{2}; 9; J_{3}; 8; 2J_{1}; 6; J_{1}⊕L_{1}; 5; L_{2}; L_{1}⊕R_{1}; J_{3}(2); 4; J_{1}⊕J_{2}; 3; J_{3}(3); 2; J_{1}⊕J_{2}(2); 1; 3J_{1}; 0; J_{1}; 4; L_{1}; J_{2}; R_{1}; 3; J_{2}(2); 1; 2J_{1}; 0; a; b]

Figure 2: Hierarchy of closures of 2 × 2 (a) and 3 × 3 (b) bundles.

# Canonical form n m c Bundle  GL_{n,m,2}-orbit SL_{n,m,2}-orbit
1 J_{1}  1 1 0  ∅
2 L_{1}  1 2 0 1  ∅
3 J_{2}  2 2 3 1
4 J_{2}(2)  2 2 1 3, 2^{∗}
5 2J_{1} 2 2 0 4 ∅
6 L_{1}+J_{1}  2 3 1 5
7 L_{2} 2 3 0 6 ∅
8 J_{3}  3 3 8 3
9 R_{1}+L_{1}  3 3 4 6^{∗}
10 J_{3}(2)  3 3 4 8, 6^{∗}
11 J_{2}+J_{1}  3 3 3 10
12 J_{3}(3)  3 3 2 10, 9, 7^{∗}
13 J_{2}(2)+J_{1}  3 3 1 12, 11
14 3J_{1}  3 3 0 13  ∅
15 2L_{1}  2 4 0 7  ∅
16 L_{1}+J_{2}  3 4 5 11
17 L_{1}+J_{2}(2)  3 4 3 16, 15, 13
18 L_{1}+2J_{1}  3 4 2 17, 14
19 L_{2}+J_{1}  3 4 1 18
20 L_{3} 3 4 0 19 ∅
21 J_{4}  4 4 15 8
22 J_{4}(2)  4 4 9 21, 16^{∗}
23 J_{3}+J_{1}  4 4 8 22
24 J_{4}(2, 2)  4 4 7 22, 17^{∗}
25 R_{1}+L_{1}+J_{1}  4 4 6 18^{∗}
26 2J_{2} 4 4 6 24, 18^{∗} ∅
27 R_{1}+L_{2}  4 4 5 25, 19
28 J_{4}(3)  4 4 5 25, 24, 19^{∗}
29 J_{3}(2)+J_{1}  4 4 4 28, 23
30 J_{2}(2)+J_{2}  4 4 4 28, 26  26
31 J_{2}+2J_{1}  4 4 3 30, 29  30
32 J_{4}(4)  4 4 3 28, 27^{∗}, 20^{∗}
33 J_{3}(3)+J_{1}  4 4 2 32, 29
34 2J_{2}(2)  4 4 2 32, 30  30
35 J_{2}(2)+2J_{1}  4 4 1 34, 33, 31  34, 31
36 4J_{1}  4 4 0 35  33  ∅
37 J_{1}+2L_{1}  3 5 2 19
38 L_{2}+L_{1}  3 5 0 37, 20
39 L_{1}+J_{3}  4 5 11 23
40 R_{1}+2L_{1}  4 5 8 37, 27
41 L_{1}+J_{3}(2)  4 5 7 39, 37, 29
42 L_{1}+J_{2}+J_{1}  4 5 6 41, 31
43 L_{1}+J_{3}(3)  4 5 5 41, 40, 38, 33
44 L_{2}+J_{2}  4 5 5 42
45 L_{1}+J_{2}(2)+J_{1}  4 5 4 43, 42, 35
46 L_{1}+3J_{1}  4 5 3 45, 36
47 L_{2}+J_{2}(2)  4 5 3 45, 44
48 L_{2}+2J_{1}  4 5 2 47, 46
49 L_{3}+J_{1}  4 5 1 48
50 L_{4} 4 5 0 49 ∅
51 J_{5}  5 5 24 21

# Canonical form n m c Bundle  GL_{n,m,2}-orb. SL_{n,m,2}-orb.
52 J_{5}(2)  5 5 16 51, 39^{∗}
53 J_{4}+J_{1}  5 5 15 52
54 J_{5}(2, 2)  5 5 12 52, 41^{∗}
55 R_{1}+L_{1}+J_{2}  5 5 11 42^{∗}
56 J_{3}+J_{2} 5 5 11 54, 42^{∗} 54
57 J_{5}(3)  5 5 10 55, 54, 44^{∗}
58 R_{1}+L_{1}+J_{2}(2) 5 5 9 55, 45^{∗}
59 J_{4}(2)+J_{1}  5 5 9 57, 53
60 J_{3}+J_{2}(2)  5 5 9 57, 56
61 R_{1}+L_{1}+2J_{1}  5 5 8 58, 46^{∗}
62 J_{3}+2J_{1} 5 5 8 60, 59 57
63 J_{5}(3, 2)  5 5 8 58, 57, 47^{∗}
64 R_{1}+L_{2}+J_{1}  5 5 7 61, 48
65 J_{4}(2, 2)+J_{1}  5 5 7 63, 59
66 J_{3}(2)+J_{2} 5 5 7 63, 61, 56, 48^{∗} 63, 56
67 R_{1}+L_{3}  5 5 6 64, 49
68 R_{2}+L_{2}  5 5 6 64^{∗}
69 J_{1}+2J_{2} 5 5 6 66, 65 ∅
70 J_{5}(4)  5 5 6 64^{∗}, 63, 49^{∗}
71 J_{4}(3)+J_{1}  5 5 5 70, 65
72 J_{3}(2)+J_{2}(2)  5 5 5 70, 66, 60
73 J_{3}(3)+J_{2}  5 5 5 70, 66
74 J_{3}(2)+2J_{1}  5 5 4 72, 71, 62  70, 62
75 J_{2}(2)+J_{2}+J_{1}  5 5 4 73, 72, 71, 69  69
76 J_{5}(5)  5 5 4 70, 68, 67^{∗}, 50^{∗}
77 J_{2}+3J_{1} 5 5 3 75, 74 73, 71 ∅
78 J_{4}(4)+J_{1}  5 5 3 76, 71
79 J_{3}(3)+J_{2}(2)  5 5 3 76, 73, 72
80 J_{3}(3)+2J_{1}  5 5 2 79, 78, 74  76, 74
81 J_{1}+2J_{2}(2)  5 5 2 79, 78, 75  75
82 J_{2}(2)+3J_{1}  5 5 1 81, 80, 77  79, 78, 77  77
83 5J_{1}  5 5 0 82  78  ∅
84 3L_{1}  3 6 0 38  ∅
85 J_{2}+2L_{1}  4 6 7 44
86 J_{2}(2)+2L_{1}  4 6 5 85, 84, 47
87 2L_{1}+2J_{1}  4 6 4 86, 48
88 L_{2}+L_{1}+J_{1}  4 6 2 87, 49
89 L_{3}+L_{1}  4 6 1 88, 50
90 2L_{2} 4 6 0 89 ∅
91 L_{1}+J_{4}  5 6 19 53
92 L_{1}+J_{4}(2)  5 6 13 91, 85, 59
93 L_{1}+J_{3}+J_{1}  5 6 12 92, 62
94 R_{1}+J_{1}+2L_{1}  5 6 11 87, 64
95 L_{1}+J_{4}(2, 2)  5 6 11 92, 86, 65
96 L_{2}+J_{3}  5 6 11 93
97 R_{2}+2L_{1}  5 6 10 94, 68
98 L_{1}+2J_{2}  5 6 10 95, 87, 69
99 R_{1}+L_{2}+L_{1}  5 6 9 94, 88, 67
100 L_{1}+J_{4}(3)  5 6 9 95, 94, 88, 71
101 L_{1}+J_{3}(2)+J_{1}  5 6 8 100, 93, 74
102 L_{1}+J_{2}(2)+J_{2}  5 6 8 100, 98, 75

# Canonical form n m c Bundle  GL_{n,m,2}-orb. SL_{n,m,2}-orb.
103 L_{1}+J_{2}+2J_{1}  5 6 7 102, 101, 77
104 L_{1}+J_{4}(4)  5 6 7 100, 99, 97, 89, 78
105 L_{2}+J_{3}(2)  5 6 7 101, 96
106 L_{1}+J_{3}(3)+J_{1}  5 6 6 104, 101, 80
107 L_{1}+2J_{2}(2)  5 6 6 104, 102, 81
108 L_{2}+J_{2}+J_{1}  5 6 6 105, 103
109 L_{1}+J_{2}(2)+2J_{1}  5 6 5 107, 106, 103, 82
110 L_{2}+J_{3}(3)  5 6 5 106, 105, 90
111 L_{3}+J_{2}  5 6 5 108
112 L_{1}+4J_{1} 5 6 4 109, 83 106, 83, 82
113 L_{2}+J_{2}(2)+J_{1}  5 6 4 110, 109, 108
114 L_{2}+3J_{1}  5 6 3 113, 112
115 L_{3}+J_{2}(2)  5 6 3 113, 111
116 L_{3}+2J_{1}  5 6 2 115, 114
117 L_{4}+J_{1}  5 6 1 116
118 L_{5} 5 6 0 117 ∅
119 J_{6}  6 6 35 51
120 J_{6}(2)  6 6 25 119, 91^{∗}
121 J_{5}+J_{1}  6 6 24 120
122 J_{6}(2, 2)  6 6 19 120, 92^{∗}
123 R_{1}+L_{1}+J_{3}  6 6 18 93^{∗}
124 J_{4}+J_{2} 6 6 18 122, 93^{∗} 122
125 J_{6}(3)  6 6 17 123, 122, 96^{∗}
126 J_{6}(2, 2, 2)  6 6 17 122, 95^{∗}
127 2R_{1}+2L_{1}  6 6 16 97^{∗}
128 J_{5}(2)+J_{1}  6 6 16 125, 121
129 J_{4}+J_{2}(2)  6 6 16 125, 124
130 2J_{3}  6 6 16 126, 98^{∗}  ∅
131 J_{4}+2J_{1}  6 6 15 129, 128  125
132 R_{1}+L_{1}+J_{3}(2)  6 6 14 123, 101^{∗}
133 R_{1}+L_{1}+J_{2}+J_{1}  6 6 13 132, 103^{∗}
134 J_{6}(3, 2)  6 6 13 132, 126, 125, 105^{∗}
135 R_{1}+L_{1}+J_{3}(3)  6 6 12 132, 127, 106^{∗}
136 R_{1}+L_{2}+J_{2}  6 6 12 133, 108
137 J_{5}(2, 2)+J_{1}  6 6 12 134, 128
138 J_{4}(2)+J_{2}  6 6 12 134, 133, 124, 108^{∗}  134, 124
139 J_{3}(2)+J_{3}  6 6 12 134, 133, 130, 108^{∗}  130
140 R_{1}+L_{1}+J_{2}(2)+J_{1} 6 6 11 135, 133, 109^{∗}
141 J_{3}+J_{2}+J_{1} 6 6 11 139, 138, 137 139
142 J_{6}(4)  6 6 11 136^{∗}, 134, 111^{∗}
143 J_{6}(3, 3)  6 6 11 135, 134, 110^{∗}
144 R_{1}+L_{1}+3J_{1}  6 6 10 140, 112^{∗}
145 R_{1}+L_{2}+J_{2}(2)  6 6 10 140, 136, 113
146 J_{5}(3)+J_{1}  6 6 10 142, 137
147 J_{4}(2)+J_{2}(2)  6 6 10 142, 138, 129
148 J_{4}(2, 2)+J_{2}  6 6 10 143, 140, 138, 113^{∗}  143, 138
149 J_{3}(3)+J_{3}  6 6 10 142, 139  139
150 R_{1}+L_{2}+2J_{1}  6 6 9 145, 144, 114
151 J_{4}(2)+2J_{1}  6 6 9 147, 146, 131  142, 131
152 J_{3}+J_{2}(2)+J_{1}  6 6 9 149, 147, 146, 141  149, 141
153 3J_{2}  6 6 9 148, 144, 114^{∗}  ∅

# Can. form  n m c Bundle  GL_{n,m,2}-orb. SL_{n,m,2}-orb.
154 J_{6}(4, 2)  6 6 9 145^{∗}, 143, 142, 115^{∗}
155 R_{1}+L_{3}+J_{1}  6 6 8 150, 116
156 R_{2}+L_{2}+J_{1}  6 6 8 150^{∗}
157 J_{3}+3J_{1} 6 6 8 152, 151 149, 146 149
158 J_{5}(3, 2)+J_{1}  6 6 8 154, 146
159 J_{4}(2, 2)+J_{2}(2) 6 6 8 154, 148, 147
160 J_{4}(3)+J_{2}  6 6 8 154, 150^{∗}, 148, 116^{∗}  154, 148
161 2J_{3}(2)  6 6 8 154, 150^{∗}, 139, 116^{∗}  139
162 R_{1}+L_{4}  6 6 7 155, 117
163 R_{2}+L_{3}  6 6 7 156, 155
164 J_{4}(2, 2)+2J_{1}  6 6 7 159, 158, 151  154, 151
165 J_{3}(2)+J_{2}+J_{1}  6 6 7 161, 160, 158, 141  161, 141
166 J_{2}(2)+2J_{2}  6 6 7 160, 159, 153  153
167 J_{6}(5)  6 6 7 156, 155^{∗}, 154, 117^{∗}
168 2J_{2}+2J_{1} 6 6 6 166, 165, 164 160, 158 ∅
169 J_{5}(4)+J_{1}  6 6 6 167, 158
170 J_{4}(3)+J_{2}(2)  6 6 6 167, 160, 159
171 J_{3}(3)+J_{3}(2) 6 6 6 167, 161, 149 161, 149
172 J_{4}(4)+J_{2}  6 6 6 167, 160
173 J_{4}(3)+2J_{1} 6 6 5 170, 169, 164 167, 164
174 J_{3}(2)+J_{2}(2)+J_{1} 6 6 5 169−171, 165, 152  171, 165, 152
175 J_{3}(3)+J_{2}+J_{1}  6 6 5 171, 172, 169, 165  171, 165
176 J_{2}+2J_{2}(2)  6 6 5 172, 170, 166  166
177 J_{6}(6)  6 6 5 167, 163^{∗}, 162^{∗}, 118^{∗}
178 J_{3}(2)+3J_{1}  6 6 4 174, 173, 157  171, 169, 157 171, 157
179 J_{2}(2)+J_{2}+2J_{1} 6 6 4 173−176, 168  168−172  168
180 J_{5}(5)+J_{1}  6 6 4 177, 169
181 J_{4}(4)+J_{2}(2)  6 6 4 177, 172, 170
182 2J_{3}(3)  6 6 4 177, 171  171
183 J_{2}+4J_{1}  6 6 3 179, 178  172, 169  ∅
184 J_{4}(4)+2J_{1}  6 6 3 181, 180, 173  177, 173
185 J_{3}(3)+J_{2}(2)+J_{1} 6 6 3 180−182, 175, 174  182, 175, 174
186 3J_{2}(2)  6 6 3 181, 176  176
187 J_{3}(3)+3J_{1}  6 6 2 185, 184, 178  182, 180, 178 182, 178
188 2J_{2}(2)+2J_{1}  6 6 2 186, 185, 184, 179 181, 180, 179 179
189 J_{2}(2)+4J_{1}  6 6 1 188, 187, 183  183, 181, 180 183
190 6J_{1}  6 6 0 189  180  ∅
191 J_{1}+3L_{1}  4 7 3 88
192 L_{2}+2L_{1}  4 7 0 191, 90
193 J_{3}+2L_{1}  5 7 14 96
194 R_{1}+3L_{1}  5 7 12 191, 99
195 J_{3}(2)+2L_{1}  5 7 10 193, 191, 105
196 J_{2}+J_{1}+2L_{1}  5 7 9 195, 108
197 J_{3}(3)+2L_{1}  5 7 8 195, 194, 192, 110
198 J_{2}(2)+J_{1}+2L_{1} 5 7 7 197, 196, 113
199 L_{2}+L_{1}+J_{2}  5 7 7 196, 111
200 3J_{1}+2L_{1}  5 7 6 198, 114
201 L_{2}+L_{1}+J_{2}(2)  5 7 5 199, 198, 115
202 L_{2}+L_{1}+2J_{1}  5 7 4 201, 200, 116
203 L_{3}+L_{1}+J_{1}  5 7 3 202, 117
204 L_{4}+L_{1}  5 7 2 203, 118

# Can. form  n m c Bundle  GL_{n,m,2}-,SL_{n,m,2}-orbit
205 J_{1}+2L_{2}  5 7 2 203
206 L_{3}+L_{2}  5 7 0 205, 204
207 L_{1}+J_{5}  6 7 29 121
208 L_{1}+J_{5}(2)  6 7 21 207, 193, 128
209 L_{1}+J_{4}+J_{1}  6 7 20 208, 131
210 L_{2}+J_{4}  6 7 19 209
211 R_{1}+J_{2}+2L_{1}  6 7 17 196, 136
212 L_{1}+J_{5}(2, 2)  6 7 17 208, 195, 137
213 L_{1}+J_{3}+J_{2}  6 7 16 212, 196, 141
214 R_{1}+J_{2}(2)+2L_{1}  6 7 15 211, 198, 145
215 L_{1}+J_{5}(3)  6 7 15 212, 211, 199, 146
216 R_{1}+2L_{1}+2J_{1}  6 7 14 214, 200, 150
217 L_{1}+J_{4}(2)+J_{1}  6 7 14 215, 209, 151
218 L_{1}+J_{3}+J_{2}(2)  6 7 14 215, 213, 152
219 R_{2}+J_{1}+2L_{1}  6 7 13 216, 156
220 L_{1}+J_{3}+2J_{1}  6 7 13 218, 217, 157
221 L_{1}+J_{5}(3, 2)  6 7 13 215, 214, 201, 158
222 L_{2}+J_{4}(2)  6 7 13 217, 210
223 R_{3}+2L_{1}  6 7 12 219, 163^{⊤}
224 R_{1}+L_{2}+L_{1}+J_{1}  6 7 12 216, 202, 155
225 L_{1}+J_{4}(2, 2)+J_{1} 6 7 12 221, 217, 164
226 L_{1}+J_{3}(2)+J_{2}  6 7 12 221, 216, 213, 202, 165
227 L_{2}+J_{3}+J_{1}  6 7 12 222, 220
228 L_{1}+J_{1}+2J_{2} 6 7 11 226, 225, 168 226, 225, 168, 166
229 R_{2}+L_{2}+L_{1}  6 7 11 224, 219, 163
230 R_{1}+L_{3}+L_{1}  6 7 11 224, 203, 162
231 L_{1}+J_{5}(4)  6 7 11 224, 221, 219, 203, 169
232 L_{2}+J_{4}(2, 2)  6 7 11 225, 222
233 L_{3}+J_{3}  6 7 11 227
234 L_{1}+J_{4}(3)+J_{1}  6 7 10 231, 225, 173
235 L_{1}+J_{3}(2)+J_{2}(2) 6 7 10 231, 226, 218, 174
236 L_{1}+J_{3}(3)+J_{2}  6 7 10 231, 226, 175
237 R_{1}+2L_{2}  6 7 10 230, 205
238 L_{2}+2J_{2}  6 7 10 232, 228
239 L_{1}+J_{3}(2)+2J_{1}  6 7 9 235, 234, 220, 178
240 L_{1}+J_{2}(2)+J_{2}+J_{1} 6 7 9 234−236, 228, 179  234−236, 228, 179, 176
241 L_{1}+J_{5}(5)  6 7 9 229−231, 223, 204, 180
242 L_{2}+J_{4}(3)  6 7 9 234, 232, 205
243 L_{1}+J_{2}+3J_{1} 6 7 8 240, 239, 183 236, 234, 183, 179, 178
244 L_{1}+J_{4}(4)+J_{1}  6 7 8 241, 234, 184
245 L_{1}+J_{3}(3)+J_{2}(2) 6 7 8 241, 236, 235, 185
246 L_{2}+J_{3}(2)+J_{1}  6 7 8 242, 239, 227
247 L_{2}+J_{2}(2)+J_{2}  6 7 8 242, 240, 238
248 L_{1}+J_{3}(3)+2J_{1}  6 7 7 245, 244, 239, 187
249 L_{1}+J_{1}+2J_{2}(2) 6 7 7 245, 244, 240, 188 245, 244, 240, 188, 186
250 L_{2}+J_{2}+2J_{1}  6 7 7 247, 246, 243
251 L_{2}+J_{4}(4)  6 7 7 244, 242, 237, 206
252 L_{3}+J_{3}(2)  6 7 7 246, 233
253 L_{1}+J_{2}(2)+3J_{1} 6 7 6 249, 248, 243, 189 243−245, 189, 188, 187
254 L_{2}+J_{3}(3)+J_{1}  6 7 6 251, 248, 246
255 L_{2}+2J_{2}(2)  6 7 6 251, 249, 247

# Canonical form n m c Bundle  GL_{n,m,2}-orb. SL_{n,m,2}-orb.
256 L_{3}+J_{2}+J_{1}  6 7 6 252, 250
257 L_{1}+5J_{1} 6 7 5 253, 190 244, 190, 189
258 L_{2}+J_{2}(2)+2J_{1}  6 7 5 255, 254, 253, 250
259 L_{3}+J_{3}(3)  6 7 5 254, 252
260 L_{4}+J_{2}  6 7 5 256
261 L_{2}+4J_{1} 6 7 4 258, 257 257, 254, 253
262 L_{3}+J_{2}(2)+J_{1}  6 7 4 259, 258, 256
263 L_{3}+3J_{1}  6 7 3 262, 261
264 L_{4}+J_{2}(2)  6 7 3 262, 260
265 L_{4}+2J_{1}  6 7 2 264, 263
266 L_{5}+J_{1}  6 7 1 265
267 L_{6} 6 7 0 266 ∅
268 J_{7}  7 7 48 119
269 J_{7}(2)  7 7 36 268, 207^{∗}
270 J_{6}+J_{1}  7 7 35 269
271 J_{7}(2, 2)  7 7 28 269, 208^{∗}
272 R_{1}+L_{1}+J_{4}  7 7 27 209^{∗}
273 J_{5}+J_{2} 7 7 27 271, 209^{∗} 271
274 J_{7}(3)  7 7 26 272, 271, 210^{∗}
275 J_{6}(2)+J_{1}  7 7 25 274, 270
276 J_{5}+J_{2}(2)  7 7 25 274, 273
277 J_{5}+2J_{1} 7 7 24 276, 275 274
278 J_{7}(2, 2, 2)  7 7 24 271, 212^{∗}
279 J_{4}+J_{3} 7 7 23 278, 213^{∗} 278
280 R_{1}+L_{1}+J_{4}(2)  7 7 21 272, 217^{∗}
281 J_{1}+2R_{1}+2L_{1}  7 7 20 219^{∗}
282 R_{1}+L_{1}+J_{3}+J_{1}  7 7 20 280, 220^{∗}
283 J_{7}(3, 2)  7 7 20 280, 278, 274, 222^{∗}
284 R_{1}+L_{1}+J_{4}(2, 2) 7 7 19 280, 225^{∗}
285 R_{1}+L_{2}+J_{3}  7 7 19 282, 227
286 J_{6}(2, 2)+J_{1}  7 7 19 283, 275
287 J_{5}(2)+J_{2}  7 7 19 283, 282, 273, 227^{∗}  283, 273
288 J_{4}+J_{3}(2)  7 7 19 283, 282, 279, 227^{∗}  283, 279
289 L_{2}+L_{1}+2R_{1}  7 7 18 281, 229, 223^{⊤}
290 R_{1}+L_{1}+2J_{2}  7 7 18 284, 228^{∗}
291 J_{4}+J_{2}+J_{1} 7 7 18 288, 287, 286 283
292 J_{7}(4)  7 7 18 285^{∗}, 283, 233^{∗}
293 J_{7}(3, 2, 2)  7 7 18 284, 283, 232^{∗}
294 R_{1}+L_{1}+J_{4}(3)  7 7 17 284, 281, 234^{∗}
295 J_{6}(3)+J_{1}  7 7 17 292, 286
296 J_{6}(2, 2, 2)+J_{1}  7 7 17 293, 286
297 J_{5}(2)+J_{2}(2)  7 7 17 292, 287, 276
298 J_{4}+J_{3}(3)  7 7 17 292, 288
299 J_{4}(2)+J_{3} 7 7 17 293, 290, 279, 238^{∗} 293, 279
300 R_{1}+L_{1}+J_{3}(2)+J_{1} 7 7 16 294, 282, 239^{∗}
301 R_{1}+L_{1}+J_{2}(2)+J_{2} 7 7 16 294, 290, 240^{∗}
302 J_{5}(2)+2J_{1}  7 7 16 297, 295, 277  292, 277
303 J_{4}+J_{2}(2)+J_{1}  7 7 16 298, 297, 295, 291  292, 291
304 J_{1}+2J_{3}  7 7 16 299, 296  ∅
305 J_{7}(3, 3)  7 7 16 294, 293, 242^{∗}
306 R_{1}+L_{1}+J_{2}+2J_{1} 7 7 15 301, 300, 243^{∗}

472 Pervouchine

# Canonical form  c Bundle  GL_{n,m,2}-orbit  SL_{n,m,2}-orb.
307 R_{1}+L_{1}+J_{4}(4)  15 294, 289^{∗}, 244^{∗}
308 R_{1}+L_{2}+J_{3}(2)  15 300, 285, 246
309 J_{4}+3J_{1}  15 303, 302  298, 295  292
310 J_{5}(2, 2)+J_{2}  15 305, 300, 287, 246^{∗}  305, 287
311 J_{4}(2, 2)+J_{3}  15 305, 301, 299, 247^{∗}  305, 299
312 R_{1}+L_{1}+J_{3}(3)+J_{1} 14 307, 300, 248^{∗}
313 R_{1}+L_{1}+2J_{2}(2)  14 307, 301, 249^{∗}
314 R_{1}+L_{2}+J_{2}+J_{1}  14 308, 306, 250
315 J_{3}+2J_{2} 14 311, 310, 306, 250^{∗} ∅
316 J_{7}(4, 2)  14 308^{∗}, 305, 292, 252^{∗}
317 R_{1}+L_{1}+J_{2}(2)+2J_{1} 13 313, 312, 306, 253^{∗}
318 R_{1}+L_{2}+J_{3}(3)  13 312, 308, 254
319 R_{1}+L_{3}+J_{2}  13 314, 256
320 R_{2}+L_{2}+J_{2}  13 314^{∗}
321 J_{6}(3, 2)+J_{1}  13 316, 296, 295
322 J_{5}(2, 2)+J_{2}(2)  13 316, 310, 297
323 J_{5}(3)+J_{2}  13 316, 314^{∗}, 310, 256^{∗}  316, 310
324 J_{4}(2)+J_{3}(2)  13 316, 314^{∗}, 299, 288, 256^{∗}  316, 299, 288
325 J_{4}(3)+J_{3}  13 316, 314^{∗}, 311, 256^{∗}  316, 311
326 R_{1}+L_{1}+4J_{1}  12 317, 257^{∗}  312, 257^{∗}, 253^{∗}
327 R_{1}+L_{2}+J_{2}(2)+J_{1} 12 318, 317, 314, 258
328 J_{5}(2, 2)+2J_{1}  12 322, 321, 302  316, 302
329 J_{4}(2)+J_{2}+J_{1}  12 324, 323, 321, 291  316, 291
330 J_{3}(2)+J_{3}+J_{1}  12 325, 324, 321, 304  304
331 J_{3}+J_{2}(2)+J_{2}  12 325, 323, 322, 315  315
332 J_{7}(5)  12 320, 319^{∗}, 316, 260^{∗}
333 J_{7}(4, 3)  12 318^{∗}, 316, 259^{∗}
334 R_{1}+L_{2}+3J_{1}  11 327, 326, 261
335 R_{1}+L_{3}+J_{2}(2)  11 327, 319, 262
336 R_{2}+L_{2}+J_{2}(2)  11 327^{∗}, 320
337 J_{3}+J_{2}+2J_{1} 11 331, 330, 329, 328 325, 323, 321 ∅
338 J_{6}(4)+J_{1}  11 332, 321
339 J_{6}(3, 3)+J_{1}  11 333, 321
340 J_{5}(3)+J_{2}(2)  11 332, 323, 322
341 J_{5}(3, 2)+J_{2} 11 333, 327^{∗}, 323, 262^{∗} 333, 323
342 J_{4}(2)+J_{3}(3)  11 332, 324, 298
343 J_{4}(2, 2)+J_{3}(2) 11 333, 327^{∗}, 324, 311, 262^{∗} 333, 324, 311
344 J_{4}(4)+J_{3}  11 332, 325
345 R_{1}+L_{3}+2J_{1}  10 335, 334, 263
346 R_{2}+L_{2}+2J_{1}  10 336, 334^{∗}
347 J_{5}(3)+2J_{1}  10 340, 338, 328  332, 328
348 J_{4}(2)+J_{2}(2)+J_{1}  10 342, 340, 338, 329, 303  332, 329, 303
349 J_{4}(2, 2)+J_{2}+J_{1}  10 343, 341, 339, 329  333, 329
350 J_{3}(3)+J_{3}+J_{1}  10 344, 342, 338, 330  330
351 J_{3}+2J_{2}(2)  10 344, 340, 331  331
352 J_{3}(2)+2J_{2}  10 343, 341, 334^{∗}, 315, 263^{∗}  315
353 J_{7}(5, 2)  10 336, 335^{∗}, 333, 332, 264^{∗}
354 R_{1}+L_{4}+J_{1}  9 345, 265
355 R_{2}+L_{3}+J_{1}  9 346, 345
356 J_{4}(2)+3J_{1} 9 348, 347, 309 342, 338, 309 332, 309
357 J_{3}+J_{2}(2)+2J_{1}  9 351, 350, 348, 347, 337 344, 340, 338, 337 337
358 J_{1}+3J_{2} 9 352, 349 341, 339, 334^{∗}, 263^{∗} ∅
359 J_{6}(4, 2)+J_{1}  9 353, 339, 338
360 J_{5}(3, 2)+J_{2}(2)  9 353, 341, 340

# Can. form  c Bundle  GL_{n,m,2}-orbit  SL_{n,m,2}-orb.
361 J_{5}(4)+J_{2}  9 353, 346, 345^{∗}, 341, 265^{∗}  353, 341
362 J_{4}(2, 2)+J_{3}(3)  9 353, 343, 342
363 J_{4}(3)+J_{3}(2) 9 353, 346, 345^{∗}, 343, 325, 265^{∗} 353, 343, 325
364 R_{1}+L_{5}  8 354, 266
365 R_{2}+L_{4}  8 355, 354
366 R_{3}+L_{3}  8 355^{∗}
367 J_{3}+4J_{1}  8 357, 356  344, 338  ∅
368 J_{5}(3, 2)+2J_{1}  8 360, 359, 347  353, 347
369 J_{4}(2, 2)+J_{2}(2)+J_{1} 8 362, 360, 359, 349, 348  353, 349, 348
370 J_{4}(3)+J_{2}+J_{1}  8 363, 361, 359, 349  353, 349
371 J_{1}+2J_{3}(2)  8 363, 359, 330  330
372 J_{3}(2)+J_{2}(2)+J_{2}  8 363, 361, 360, 352, 331  352, 331
373 J_{3}(3)+2J_{2}  8 362, 361, 352  352
374 J_{7}(6)  8 355^{∗}, 354^{∗}, 353, 266^{∗}
375 J_{4}(2, 2)+3J_{1}  7 369, 368, 356  362, 359, 356  356, 353
376 J_{2}(2)+J_{1}+2J_{2}  7 373, 372, 370, 369, 358  361, 360, 359, 358 358
377 J_{3}(2)+J_{2}+2J_{1}  7 372, 371, 370, 368, 337  363, 361, 359, 337 337
378 J_{6}(5)+J_{1}  7 374, 359
379 J_{5}(4)+J_{2}(2)  7 374, 361, 360
380 J_{5}(5)+J_{2}  7 374, 361
381 J_{4}(3)+J_{3}(3)  7 374, 363, 362
382 J_{4}(4)+J_{3}(2)  7 374, 363, 344
383 3J_{1}+2J_{2}  6 376, 377, 375  361, 359  ∅
384 J_{4}(4)+J_{2}+J_{1}  6 382, 380, 378, 370  374, 370
385 J_{4}(3)+J_{2}(2)+J_{1}  6 381, 379, 378, 370, 369  374, 370, 369
386 J_{5}(4)+2J_{1}  6 379, 378, 368  374, 368
387 J_{3}(3)+J_{3}(2)+J_{1}  6 382, 381, 378, 371, 350  371, 350
388 J_{3}(2)+2J_{2}(2)  6 382, 379, 372, 351  372, 351
389 J_{3}(3)+J_{2}(2)+J_{2}  6 381, 380, 379, 373, 372  373, 372
390 J_{7}(7)  6 374, 366, 365^{∗}, 364^{∗}, 267^{∗}
391 J_{4}(3)+3J_{1} 5 385, 386, 375 381, 378, 375 375, 374
392 J_{3}(2)+J_{2}(2)+2J_{1} 5 385−388, 377, 357  382, 377−379, 357 377, 357
393 J_{3}(3)+J_{2}+2J_{1}  5 389, 387, 384, 386, 377  381, 380, 378, 377 377
394 J_{2}+J_{1}+2J_{2}(2)  5 389, 388, 384, 385, 376  380, 379, 378, 376 376
395 J_{6}(6)+J_{1}  5 390, 378
396 J_{5}(5)+J_{2}(2)  5 390, 380, 379
397 J_{4}(4)+J_{3}(3)  5 390, 382, 381
398 J_{3}(2)+4J_{1}  4 392, 391, 367  382, 378, 367  367
399 J_{2}(2)+J_{2}+3J_{1}  4 394, 393, 392, 391, 383  383, 380, 379, 378 383
400 J_{5}(5)+2J_{1}  4 396, 395, 386  390, 386
401 J_{4}(4)+J_{2}(2)+J_{1}  4 397, 396, 395, 384, 385  390, 384, 385
402 J_{1}+2J_{3}(3)  4 397, 395, 387  387
403 J_{3}(3)+2J_{2}(2)  4 397, 396, 389, 388  389, 388
404 J_{2}+5J_{1}  3 399, 398  380, 378  ∅
405 J_{4}(4)+3J_{1}  3 401, 400, 391  397, 395, 391  391, 390
406 J_{3}(3)+J_{2}(2)+2J_{1} 3 400−403, 393, 392  395−397, 393, 392 393, 392
407 J_{1}+3J_{2}(2)  3 403, 401, 394  396, 395, 394  394
408 J_{3}(3)+4J_{1}  2 406, 405, 398  398, 397, 395  398
409 3J_{1}+2J_{2}(2)  2 407, 406, 405, 399  399, 396, 395  399
410 J_{2}(2)+5J_{1}  1 409, 408, 404  404, 396, 395  404
411 7J_{1}  0 410  395  ∅

Table 1: Bundles and orbits of matrix pencils.

[figure: 8; 16; 21; 15; 7; 13; 15; 12; 11; 11; 16; 10; 13; 22; 9; 17; 14; 23; 8; 18; 24; 7; 19; 25; 26; 6; 20; 27; 28; 5; 29; 30; 4; 31; 32; 3; 33; 34; 2; 35; 1; 36; 0; 21; 25; 51; 24; 23; 18; 19; 39; 17; 20; 37; 52; 16; 27; 53; 15; 38; 40; 29; 14; 41; 31; 13; 42; 33; 54; 12; 43; 44; 35; 55; 56; 11; 45; 36; 57; 10; 46; 47; 58; 59; 60; 9; 48; 61; 62; 63; 8; 49; 64; 65; 66; 7; 50; 67; 68; 69; 70; 6; 71; 72; 73; 5; 74; 75; 76; 4; 77; 78; 79; 3; 80; 81; 2; 82; 1; 83; 0; a; b]

Figure 3: Hierarchy of closures of 4 × 4 (a) and 5 × 5 (b) bundles.

[figure: 91; 19; 84; 18; 85; 15; 86; 92; 13; 87; 93; 12; 94; 95; 96; 11; 88; 97; 98; 10; 89; 99; 100; 9; 90; 101; 102; 8; 103; 104; 105; 7; 106; 107; 108; 6; 109; 110; 111; 5; 112; 113; 4; 114; 115; 3; 116; 2; 117; 1; 118; 0; 38; 44; 47; 48; 49; 50; 53; 59; 62; 64; 65; 68; 69; 67; 71; 74; 75; 77; 78; 80; 81; 82; 83]

Figure 4: Hierarchy of closures of 5 × 6 bundles.

[figure: 150; 151; 152; 153; 154; 9; 155; 156; 157; 158; 159; 160; 161; 8; 162; 163; 164; 165; 166; 167; 7; 168; 169; 170; 171; 172; 6; 173; 174; 175; 176; 177; 5; 178; 179; 180; 181; 182; 4; 183; 184; 185; 186; 3; 187; 188; 2; 189; 1; 190; 0; 114; 116; 116; 116; 117; 117; 118]

Figure 5: Hierarchy of closures of 6 × 6 bundles.

[figure: 119; 35; 120; 25; 121; 24; 122; 19; 123; 124; 18; 125; 126; 17; 127; 128; 129; 130; 16; 131; 15; 132; 14; 133; 134; 13; 136; 135; 137; 138; 139; 12; 140; 141; 142; 143; 11; 145; 144; 146; 147; 148; 149; 10; 51; 91; 92; 93; 93; 96; 95; 97; 98; 101; 103; 105; 106; 108; 108; 108; 109; 111; 110; 112; 113; 113; 114; 115]

Figure 5: (continued)

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Dmitri D. Pervouchine Department of Mathematics and Statistics Boston University 111 Cummington Street Boston MA 02215, USA dp@bu.edu

Received April 30, 2003