On the uniqueness of the Lyapunov scaling factors. (English) Zbl 0627.15007
A matrix \(A\in R^{n,n}\) is Lyapunov diagonally stable or semistable if there exists a positive diagonal matrix D satisfying \(AD+DA^ T>0\) or \(AD+DA^ T\geq 0\), respectively. The matrix D is called a Lyapunov scaling factor of A. A unique Lyapunov scaling factor of A means uniqueness up to multiplication by a scalar. A matrix A is Lyapunov diagonally near stable if it is Lyapunov diagonally semistable but not Lyapunov diagonally stable. If A is a Lyapunov diagonally near stable irreducible P-matrix (i.e. matrix having all of principal minors positive), then A has a unique Lyapunov scaling factor as conjectured by D. Hershkowitz and H. Schneider [Linear Multilinear Algebra 17, 203-226 (1985; Zbl 0593.15016)]. This conjecture is proved for \(n\leq 3\) and a counterexample is given for \(n\geq 4\).
Reviewer: L.Bakule
MSC:
| 15A45 | Miscellaneous inequalities involving matrices |
| 15A12 | Conditioning of matrices |
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
| 15A03 | Vector spaces, linear dependence, rank, lineability |
Keywords:
Lyapunov diagonal stability; Lyapunov scaling factor; irreducible P- matrix; conjecture; counterexampleCitations:
Zbl 0593.15016References:
| [1] | Barker, G. P.; Berman, A.; Plemmons, R. J., Positive diagonal solutions to the Lyapunov equations, Linear and Multilinear Algebra, 5, 249-266 (1978) · Zbl 0385.15006 |
| [2] | Berman, A., Cones, Matrices and Mathematical Programming (1973), Springer · Zbl 0256.90002 |
| [3] | Berman, A.; Hershkowitz, D., Matrix diagonal stability and its implications, SIAM J. Algebraic Discrete Methods, 4, 377-382 (1983) · Zbl 0547.15009 |
| [4] | Berman, A.; Varga, R. S.; Ward, R. C., ALPS: Matrices with nonpositive off-diagonal entries, Linear Algebra Appl., 21, 233-244 (1978) · Zbl 0401.15018 |
| [5] | Cross, G. W., Three types of matrix stability, Linear Algebra Appl., 20, 253-263 (1978) · Zbl 0376.15007 |
| [6] | Gantmacher, F. R., The Theory of Matrices, Vol. II (1959), Chelsea · Zbl 0085.01001 |
| [7] | Hershkowitz, D.; Schneider, H., Scalings of vector spaces and the uniqueness of Lyapunov scaling factors, Linear and Multilinear Algebra, 17, 203-226 (1985) · Zbl 0593.15016 |
| [8] | Hershkowitz, D.; Schneider, H., Semistability factors and semifactors, Contemp. Math., 47, 203-216 (1985) · Zbl 0576.15012 |
| [9] | Lyapunov, A., Problème générade de la stabilité du mouvement, Comm. Math. Soc. Kharkow. Comm. Math. Soc. Kharkow, Ann. Math. Stud. 17 (1947), Princeton U.P: Princeton U.P Princeton, republished in · Zbl 0031.18403 |
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