PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 81(95) (2007), 79–83 DOI 102298/PIM0795079A

ON THE EIGENVALUES OF SOME CLASS OF PSEUDO-LINEAR TRANSFORMATIONS

Milica An - delić

Abstract. We develop a connection between the eigenvalues of a class of pseudo-linear transformation over a field K and the eigenvalues of a certain linear transformation. We give a new criterion for this class to be diagonalizable over algebraically closed field.

1. Introduction

This work was inspired by [6] – a brief study of (σ, δ) pseudo-linear transformations together with their relations with evaluations of skew polynomial rings. It contains the necessary and sufficient conditions for the algebraic pseudo-linear transformations to be diagonalizable, as well.

If K is a division ring and V K-vector space by a (σ, δ) pseudo-linear transformation we call an additive map T : V → V such that

T (αv) = σ(α)T (v) + δ(α)v, α ∈ K, v ∈ V,

where σ is an automorphism of K and δ is a left σ-derivation i.e., δ is an additive endomorphism of K such that

δ(ab) = σ(a)δ(b) + δ(a)b, a, b ∈ K.

Throughout this paper, we will assume that K is a field, δ = 0 and σ is an automorphism of K of finite order. The reason why we switched to this case is the connection that can be developed between the eigenvalues of this class of pseudolinear transformations and the eigenvalues of certain linear transformations. The use of linear transformations enables us to use the Cayley–Hamilton theorem which in a pseudo-linear setting does not hold.

The paper is organized as follows. In Section 2, we mention some results from [5, 6] in order to make the paper more self-contained. All of them are modified due to the restrictions we have made. In Section 3, the main result is presented.

2000 Mathematics Subject Classification: 15A24, 15A04, 15A18, 16S36.

Key words and phrases: Skew polynomials; pseudo-linear transformations.

2. Preliminaries

Let K be a field and σ ∈ Aut(K). A skew polynomial ring (also called Ore extension), K[t; σ] consists of polynomials ∑_{i=0}^{n} a_{i}t^{i}, a_{i} ∈ K which are added in the usual way but are multiplied according to the following commutation rule

ta = σ(a)t, a ∈ K.

For any c ∈ K^{∗} element σ(c)ac^{−1} is called the σ-conjugate of a (by c). The set {σ(c)ac^{−1} | c ∈ K^{∗}} is called the σ-conjugacy class of a.

The evaluation f(a) of a polynomial f(t) ∈ K[t; σ] at some element a ∈ K is the remainder of f(t) = ∑_{i=0}^{n} a_{i}t^{i} divided on the right by t − a. It is easy to show by induction that

f(a) = ∑_{i=0}^{n} a_{i}N_{i}(a)

where the maps N_{i} are defined by induction in the following way: For any a ∈ K

N_{0}(a) = 1 and N_{i+1}(a) = σ(N_{i}(a))a,

which leads to

N_{k}(a) = σ^{k−1}(a)σ^{k−2}(a) … σ(a)a (k ∈ ℕ).

We define f(A) for A ∈ M_{n}(K) in a similar way f(A) = ∑_{i=0}^{n} a_{i}N_{i}(A), where σ has been extended to M_{n}(K) in the natural way.

Let V be a K vector space. A σ-pseudo-linear transformation of V is an additive map T : V → V such that T (αv) = σ(α)T (v), α ∈ K. We will use the abbreviation σ-PLT for a pseudo-linear transformation with respect to the automorphism σ. A vector v ∈ V ∖ {0} is an eigenvector of σ-PLT with the corresponding eigenvalue λ ∈ K iff T (v) = λv.

If V is finite-dimensional and e = [e_{1}, …, e_{n}] is a basis of V let us write T (e_{j}) = ∑_{i=1}^{n} a_{ij}e_{i}, a_{ij} ∈ K or in the matrix notation T e = eA where A = [a_{ij}] ∈ M_{n}(K). The matrix A will be denoted [T]_{e}. The equality [f(T)]_{e} = f([T]_{e}) holds as well [6, Proposition 2.13] for any polynomial f(t) ∈ K[t, σ]. Also if v is an eigenvector of σ-PLT T with an eigenvalue λ ∈ K, then [T]_{e}σ(v_{e})^{T} = λv_{e}^{T} , where v_{e} denotes the coordinates of the vector v with respect to the basis e [3].

A σ-PLT transformation T is algebraic if there exist m ∈ ℕ, a_{0}, a_{1}, …, a_{m} ∈ K, a_{m} ≠ 0 such that

a_{m}T^{m} + ⋯ + a_{1}T + a_{0}I = 0.

In the case T is algebraic σ-PLT on V and μ_{T} ∈ K[t; σ] is its minimal polynomial, λ ∈ K is an eigenvalue for T if and only if t − λ divides on the right (left) the polynomial μ_{T} in K[t; σ] [6, Proposition 4.5].

We will also use the notion of a Wedderburn polynomial. For f ∈ K[t; σ], let

V (f) := {a ∈ K | f(a) = 0}.

A (monic) polynomial is said to be Wedderburn if f = μ_{V (f)} i.e., f is equal to the minimal polynomial of V (f)-set of its roots [5].

3. General results

Let K be a field, σ ∈ Aut(K) of an order k i.e., σ ≠ id_{K} and k is the least nonnegative integer such that σ^{k} = id_{K}. If T is σ-PLT on a vector space V over K, then T^{k} is a linear transformation of V since it is additive and

T^{k}(αv) = σ^{k}(α)T^{k}(v) = αT^{k}(v), α ∈ K.

Therefore, if V is a finite-dimensional vector space there exist m ∈ ℕ, a_{0}, …, a_{m} ∈ K, a_{m} ≠ 0 such that

a_{m}(T^{k})^{m} + ⋯ + a_{1}T^{k} + a_{0}I = 0,

which means that σ-PLT T is algebraic. We will denote its minimal polynomial with μ_{T}. This polynomial is invariant in K[t; σ] and also the right factor of the polynomial φ_{T^{k}}(t^{k}), where φ_{T^{k}} denotes the characteristic polynomial of T^{k}. What we want is to find relations between eigenvalues of linear transformation T^{k} and σ-PLT T.

Theorem 3.1. Let T be σ-PLT on a finite dimensional vector space V over field K and σ ∈ Aut(K) of an order k. An element λ ∈ K is the eigenvalue of T iff N_{k}(λ) is the eigenvalue of T^{k}.

Proof. Let v ∈ V ∖ {0} be such that T (v) = λv. Then

T^{k}(v) = T^{k−1}(λv) = σ^{k−1}(λ)T^{k−1}(v) = ⋯
= σ^{k−1}(λ) ⋯ σ(λ)λv = N_{k}(λ)v.

The polynomial h(t) = t^{k}−N_{k}(λ) is a Wedderburn polynomial, since it is a minimal polynomial of the set Γ = {σ(c)λc^{−1} | c ∈ K^{∗}}. For any c ∈ K^{∗}, we have

N_{k}(σ(c)λc^{−1}) = σ^{k}(c)N_{k}(λ)c^{−1} = N_{k}(λ).

This shows that h vanishes on Γ. Let f(t) = ∑_{i=1}^{m} a_{i}t^{i} be the monic minimal polynomial of Γ. Then m = deg f ⩽ k, and a_{0} ≠ 0. Let d ∈ K^{∗}. For any e ∈ Γ, we have 0 = ∑_{i=0}^{m} a_{i}σ^{i}(d)N_{i}(e)d^{−1}. Thus, Γ satisfies the polynomial ∑_{i=0}^{m} a_{i}σ^{i}(d)t^{i}. By the uniqueness of the minimal polynomial, we must have σ^{m}(d)a_{i} = a_{i}σ^{i}(d) for every i. Since a_{0} ≠ 0, this implies that σ^{m} = id_{K}. Therefore, we have m = k and f(t) = t^{k} − N_{k}(λ).

We can write t^{k} − N_{k}(λ) = (t − λ_{k})(t − λ_{k−1}) ⋯ (t − λ_{1}), where λ_{1}, …, λ_{k} are σ-conjugated to λ [5, Theorem 5.1], [3, Lemma 5]. This gives us

T^{k} − N_{k}(λ) id_{K} = (T − λ_{k} id_{K})(T − λ_{k−1} id_{K}) ⋯ (T − λ_{1} id_{K}).

We can conclude that if there exists 0 ≠ v ∈ V such that (T^{k} − N_{k}(λ) id_{K})(v) = 0, then there exists l ∈ {1, …, k} and 0 ≠ u ∈ V such that (T − λ_{l} id_{K})(u) = 0. Since that λ_{l} is σ-conjugated to λ, there exists a ∈ K^{∗} such that λ_{l} = σ(a)λa^{−1}. Then for u_{0} = a^{−1}u we obtain

T (u_{0}) = T (a^{−1}u) = σ(a^{−1})T (u) = σ(a^{−1})σ(a)λa^{−1}u = λu_{0}

i.e., λ is an eigenvalue for T, as desired. □

According to [6, Theorem 4.6] the minimal polynomial of an algebraic σ-PLT T has roots in at most n = dim V σ-conjugacy classes. Moreover [6, Proposition 4.3] shows that the set

Γ_{T} = {α ∈ K | T (v) = αv}

is closed by σ-conjugations. We can write Γ_{T} = Γ_{1} ∪ ⋯ ∪ Γ_{r}, where Γ_{i} = {σ(c)λ_{i}c^{−1} | c ∈ K^{∗}} and r ⩽ n i.e., we can see Γ_{T} as a disjoint union of different σ-conjugacy classes of eigenvalues of T.

Theorem 3.2. A σ-PLT T on a finite dimensional vector space V over algebraically closed field K, σ ∈ Aut(K) of an order k, is diagonalizable iff μ_{T^{k}}(t) = ∏_{i=1,λ_{i}≠λ_{j}}^{r}(t − N_{k}(λ_{i})), where r is the number of σ-conjugacy classes containing eiganvalues of T.

Proof. [6, Theorem 4.9] says that algebraic σ-PLT T is diagonalizable iff μ_{T} = ∏_{i=1}^{r} μ_{Γ_{i}} where μ_{Γ_{i}} stands for the minimal polynomial of the set Γ_{i} = {σ(c)λ_{i}c^{−1} | c ∈ K^{∗}} of the eigenvalues of T σ-conjugated to λ_{i} i.e., μ_{T} (t) = ∏_{i=1}^{r}(t^{k} −N_{k}(λ_{i})), since that μ_{Γ_{i}}(t) = t^{k} − N_{k}(λ_{i}). The polynomial ∏_{i=1}^{r}(t − N_{k}(λ_{i})) vanishes at T^{k} and is its minimal polynomial, as well. Otherwise, we would get the polynomial of degree less than deg μ_{T} vanishing at T. Thus, μ_{T^{k}}(t) = ∏_{i=1}^{r}(t − N_{k}(λ_{i})) as desired. □

Example 3.1. Let

A = [\begin{matrix}i + 1 & 1 \\ −1 & −i \end{matrix}] ∈ M_{2}(ℂ),

σ ∈ Aut(ℂ), τ(x) = x̄, be complex conjugation and T (X) = X̄A − AX̄. In this case, σ is the automorphism of ℂ of order k = 2.

First, we determine the matrix [T]_{e}, where e is the canonical base of M_{2}(ℂ),

P = [T]_{e} = [\begin{matrix}0 & −1 & −1 & 0 \\ 1 & −2i − 1 & 0 & −1 \\ 1 & 0 & 2i + 1 & −1 \\ 0 & 1 & 1 & 0 \end{matrix}] ,

then the matrix N_{2}(P) becomes

N_{2}(P) = P̄P = [\begin{matrix}−2 & 2i + 1 & −2i − 1 & 2 \\ 2i − 1 & 3 & −2 & −2i + 1 \\ −2i + 1 & −2 & 3 & 2i − 1 \\ 2 & −2i − 1 & 2i + 1 & −2 \end{matrix}] .

Next, we calculate the characteristic polynomial φ_{N_{2}(P)}. In this case we have

φ_{T^{2}}(t) = t^{2}(t − 1)^{2}

and μ_{T^{2}}(t) = t(t − 1). According to Theorem 2 σ-PLT is diagonalizable.

All the eigenvectors of T for the eigenvalue λ = 1 belong to the set U = ker(T^{2} − I) which has the basis [C, D], where

C = [\begin{matrix}−1 & 0 \\ 1 − 2i & 1 \end{matrix}] , D = [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] .

The vectors C + T (C), D + T (D) are the eigenvectors of T for λ = 1, and c^{−1}(C + T (C)), c^{−1}(D + T (D)) are the eigenvectors of T for λ = i, where c is any complex number which satisfies 1 = c̄ic^{−1}. Similarly, we get the basis [E, F] of W = ker T^{2}

E = [\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}] , F = [\begin{matrix}−2i − 1 & −1 \\ 1 & 0 \end{matrix}]

and one basis of ker T [E, T (F)].

Finally, we get that with respect to the basis [C + T (C), D + T (D), E, T (F)] σ-PLT has the matrix diag(1, 1, 0, 0). We can also get [T]_{f} = diag(i, i, 0, 0) with respect to the basis f = [c^{−1}(C + T (C)), c^{−1}(D + T (D)), E, T (F)], for c = 1 + i.

Bibliography

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[2] D. Burde, On the matrix equation XA−AX = X^{p}, Linear Algebra Appl. 404 (2005), 147–165.

[3] P. M. Cohn, Free Rings and Their Relations, Academic Press, 1985.

[4] T. Y. Lam, A general theory of Vandermonde matrices, Expo. Math. 4:3 (1986), 193–215.

[5] T. Y. Lam, A. Leroy, Wedderburn polynomials over divisoin rings I, J. Pure Appl. Algebra 186:1 (2004), 43–76.

[6] A. Leroy, Pseudo linear transformation and evaluation in Ore extensions, Bull. Belg. Math. Soc. Simon Stevin 2:3 (1995), 321–347.

Matematički fakultet

11000 Beograd

Serbia

    milica@matf.bg.ac.yu