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Quasidirect decompositions of Hankel and Toeplitz matrices. (English) Zbl 0548.15022

An \(n\times n\) matrix \(A=[a_{ij}]\) is called a Hankel matrix if the values of its entries are constant along the diagonals \(i+j=m\) for \(m=2,3,...,2n\). If \(A=B+C\) then A is said to be a quasidirect sum of B and C if for some invertible P and Q and suitable square blocks \(B_ 0\) and \(C_ 0\) we have \(B=P\quad diag(B_ 0,0)Q, C=P\quad diag(0,C_ 0)Q\) and \(A=P\quad diag(B_ 0,C_ 0)Q.\) It is easily shown that A is a quasidirect sum if and only if its rank \(r(A)=r(B)+r(C).\) The Hankel matrix A is called H-indecomposable if it cannot be written as a quasidirect sum of two nonzero Hankel matrices.
The main result of the paper is to prove that, over an algebraically closed field, every Hankel matrix is a quasidirect sum of H- indecomposable matrices and this decomposition is unique if and only if the matrix is nonsingular (Theorem 2.23). Furthermore all H- indecomposable matrices are described, and some weaker results are proved in the case the field is not algebraically closed. The author notes that, since there is a simple linear transformation from the space of Hankel matrices to the space of Toeplitz matrices, analogous results for Toeplitz matrices also hold.
Reviewer: J.D.Dixon

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
15A21 Canonical forms, reductions, classification
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References:

[1] Fiedler, M., Remarks on the Schur complement, Linear Algebra Appl., 39, 189-196 (1981) · Zbl 0465.15004
[2] Gantmacher, F. R., The Theory of Matrices (1959), Chelsea: Chelsea New York · Zbl 0085.01001
[3] Haynsworth, E. V., Determination of the inertia of a partitioned Hermitian matrix, Linear Algebra Appl., 1, 73-81 (1968) · Zbl 0155.06304
[4] Lancaster, P., Theory of Matrices (1969), Academic: Academic New York · Zbl 0186.05301
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