We introduce matrix-weighted weak Triebel-Lizorkin spaces and establish the equivalence between the corresponding weak discrete sequence spaces. In the scalar unweighted case, we first prove the boundedness of almost diagonal operators on the weak discrete Triebel-Lizorkin space and then extend this result to the matrix-weighted setting. Furthermore, we provide a characterization of these spaces in terms of molecules. Additionally, we demonstrate the equivalence between the continuous function spaces defined via a sequence of reducing operators and those defined directly by matrix weights. These results ultimately establish a complete connection between matrix-weighted weak Triebel-Lizorkin spaces and their discrete or sequence space analogues. Within this framework, we develop several characterizations of matrix-weighted weak Triebel-Lizorkin spaces: First, using the doubling property of matrix weights and the Fefferman-Stein inequality, we obtain the characterization of matrix-weighted weak Triebel-Lizorkin spaces in terms of the Peetre maximal function. Second, combining the Peetre maximal function with the Fefferman-Stein inequality, we derive the Lusin area function characterization of matrix-weighted weak Triebel-Lizorkin spaces. Third, we utilize reducing operators and the Fefferman-Stein inequality to provide the Littlewood-Paley g*_λ-function characterization of matrix-weighted weak Triebel-Lizorkin spaces. Finally, as an application, the boundedness of the classical Calderon-Zygmund operator on these spaces is obtained.

The work is supported by the National Natural Science Foundation of China (Grant No. 12161022) and the Science and Technology Project of Guangxi (Guike AD23023002).