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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.

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 |

Full Text:
DOI

### 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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