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Zbl 0936.15014
Johnson, Russell; Tesi, Alberto
On the $D$-stability problem for real matrices. (English)
[J] Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 2, No.2, 299-314 (1999). ISSN 0392-4041

A real $n\times n$ matrix $A$ is $D$-stable if for any diagonal matrix $D$ with positive diagonal elements the eigenvalues of $DA$ all lie in the left half-plane. This paper considers conditions which together with the stability of $A$ imply $D$-stability of $A$. An orthant condition sufficient for $D$-stability as well as a related analytic condition related to {\it D. Carlson}'s condition [J. Res. Natl. Bur. Stand., Sect. B 78, 1--2 (1974; Zbl 0281.15020)] are given. The latter is sharpened. It is shown that the set ${\cal D}$ of $D$-stable $n\times n$ matrices is the complement of a set describable by finitely many polynomial equations and inequalities; consequently, this set has finitely many connected components. Robust $D$-stability is discussed for $4\times 4$ matrices.
[E. Kreyszig (Ottawa)]

MSC 2000:: *15A42 Inequalities involving eigenvalues and eigenvectors

Keywords: polynomial equations; Carlson's condition; robustness; $D$-stability; eigenvalues; orthant condition; inequalities

Citations: Zbl 0281.15020

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