Structured Mapping Problems Associated With Multilevel Block g‐Toeplitz Matrices

ABSTRACT

In this article, we investigate the multilevel block ‐Toeplitz matrix with , where each level possesses a block ‐Toeplitz structure and the innermost level remains unstructured. The necessary and sufficient conditions for a matrix to be a multilevel block ‐Toeplitz matrix are derived. Next, matrix expansion techniques are utilized to facilitate the application of Vandermonde factorization in solving the structured mapping problems associated with this kind of matrices. We consider the above problems whose coefficient matrices possess the multilevel block ‐Toeplitz structure, where the solutions belong to the Jordan algebra, the Lie algebra and the automorphism group. In particular, the analytical expressions for the solutions to these problems are derived. Furthermore, a complete analytical solution is provided for the open problem proposed by Mackey et al. [SIAM J. Matrix Anal. Appl., 2008, 29(4), 1389‐1410], which is a kind of structured mapping problem with unstructured coefficient matrices. Finally, several algorithms are proposed, and corresponding numerical examples are presented to verify the rationality and feasibility of the results obtained in this study.