Characterization and properties of r-Toeplitz matrices
If a block matrix \(B=[B_{ij}]\) of order \(n=mr\) whose elements are matrices \(B_{ij}\) of order r is a block-Toeplitz matrix, i.e. if \(B_{i+1,j+1}=B_{ij}\) \((i,j=1,...,m-1)\), then for the individual elements \(b_{ij}\) of B we have \(b_{i+r,j+r}=b_{ij}\) \((i,j=1,2,...,(m-1)r)\). A matrix \(A=[a_{ij}]\) of arbitrary order n is called an rT (r-Toeplitz) matrix if \(a_{i+r,j+r}=a_{ij}\) \((i,j=1,2,...,n-r)\). An rT matrix is completely determined by its first r rows and columns and can be obtained from a block Toeplitz matrix by removing a bottom border and a right-hand border. A Toeplitz matrix is a 1T matrix, and a CT (conjugate Toeplitz) matrix [studied earlier by the authors in Linear Multilinear Algebra 14, 45-65 (1983; Zbl 0536.15011) and in Linear Algebra Appl. 61, 253-275 (1984; Zbl 0554.15012)] is a 2T matrix. It is shown here that some known properties of Toeplitz and CT matrices extend to rT matrices. In particular, a necessary and sufficient condition is given, in terms of a linear matrix equation, for a nonsingular matrix A to be an rT matrix (Theorem 2.1). From this it follows that the elements of the inverse \(A^{-1}\) of an rT matrix A are completely determined by those in its first and last r rows and columns (Theorem 2.2). In addition, some results are presented for cases when the inverse \(A^{-1}\) of an rT matrix A has certain banded forms (Theorems 3.1 and 3.2). Further properties of rT matrices will be developed in a subsequent paper of the authors (to appear in Linear Algebra Appl.).
scientific article; zbMATH DE number 4010699
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