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Stability of symmetrizable interval matrices. (English) Zbl 0957.15017
A square matrix \(A\) is called symmetrizable if \(MA=A^TM\) for a positive definite matrix \(M\). The author first studies interval matrices \(A^I=[\underline A,\overline A]\) where both \(\underline A\) and \(\overline A\) are symmetrizable by a diagonal matrix \(M\) with positive diagonal entries. Following J. Rohn [SIAM J. Matrix Anal. Appl. 15, No. 1, 175-184 (1994; Zbl 0796.65065)] it is shown that checking the Hurwitz stability of such an interval matrix can be done by checking the stability of a finite subset of matrices contained therein. Second, \(A^I\) is called \(M\)-Hurwitz (\(M\)-Schur) stable if each \(A\in A^I\) symmetrizable by \(M\) is Hurwitz (Schur) stable. It is shown that \(A^I\) is \(M\)-Hurwitz stable if and only if the interval matrix \(M^{1/2}A^I(M^{1/2})^{-1}\) is Hurwitz stable, and a similar characterization is proved for the \(M\)-Schur stability as well.
Reviewer: Jiri Rohn (Praha)
15A42 Inequalities involving eigenvalues and eigenvectors
65G30 Interval and finite arithmetic