The \(\alpha\)-scalar diagonal stability of block matrices. (English) Zbl 1107.15014
The paper deals with diagonal stability of block matrices. The author generalizes two results: the characterization of diagonal stability via Hadamard products [cf. J. F. B. M. Kraaijevanger, ibid. 151, 245–254 (1991; Zbl 0724.15014)] and the block matrix version of the closure of the positive definite matrices under Hadamard multiplication.
Given a real matrix \(A\), of size \(m \times m\), the main result of this paper establishes the equivalence between the following statements: (1) \(A\) is diagonally stable. (2) \(A \circ Q\) is diagonally stable, for every \(m \times m\) symmetric, positive semidefinite block matrix \(Q\), with diagonal blocks positive definite matrices. (3) \(A \circ Q\) is P-matrix, for every \(m \times m\) symmetric, positive semidefinite block matrix \(Q\), with diagonal blocks positive definite matrices.
Finally, the author restates her generalizations in terms of \(P^{\alpha}\)-matrices and \(\alpha\)-scalar diagonally stable matrices.
Given a real matrix \(A\), of size \(m \times m\), the main result of this paper establishes the equivalence between the following statements: (1) \(A\) is diagonally stable. (2) \(A \circ Q\) is diagonally stable, for every \(m \times m\) symmetric, positive semidefinite block matrix \(Q\), with diagonal blocks positive definite matrices. (3) \(A \circ Q\) is P-matrix, for every \(m \times m\) symmetric, positive semidefinite block matrix \(Q\), with diagonal blocks positive definite matrices.
Finally, the author restates her generalizations in terms of \(P^{\alpha}\)-matrices and \(\alpha\)-scalar diagonally stable matrices.
Reviewer: Juan Ramon Torregrosa Sanchez (Valencia)
MSC:
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
Keywords:
\(\alpha\)-scalar diagonal stability; \(P^{\alpha}\)-matrix; \(P\)-matrix; Hadamard product; positive definite matrixCitations:
Zbl 0724.15014References:
| [1] | Barker, G. P.; Bermann, A.; Plemmons, R. J., Positive diagonal solutions to the Lyapunov equations, Linear Multilinear Algebra, 5, 249-256 (1978) · Zbl 0385.15006 |
| [2] | Kaszkurewicz, E.; Bhaya, A., Matrix Diagonal Stability in Systems and Computation (2000), Birkhäuser: Birkhäuser Boston · Zbl 0951.93058 |
| [3] | Kraaijevanger, J. F.B. M., A characterization of Lyapunov diagonal stability using Hadamard products, Linear Algebra Appl., 151, 245-254 (1991) · Zbl 0724.15014 |
| [4] | Hershkowitz, D.; Mashal, N., \(P^α\)-matrices and Lyapunov scalar stability, Electron. J. Linear Algebra, 4, 39-47 (1998) · Zbl 0912.15021 |
| [5] | Horn, R. A.; Johnson, Ch. R., Topics in Matrix Analysis (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0729.15001 |
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