Positive-definiteness of interval families of symmetric matrices (Q1883821)

The main result, Theorem 1, which states that checking positive definiteness of an interval family of \(n\)-by-\(n\) symmetric matrices is equivalent to checking positive definiteness of \(2^{n-1}\) vertex matrices from the family, was already published in [\textit{D. Hertz}, ``The extreme eigenvalues and stability of real symmetric interval matrices'', IEEE Trans. Autom. Control 37, 532--535 (1992)]. In general, the cardinality of the set of matrices to be tested is exponential in \(n\), and it has been shown in [\textit{J. Rohn}, ``Checking positive definiteness or stability of symmetric interval matrices is NP-hard'', Commentat. Math. Univ. Carol. 35, 795--797 (1994; Zbl 0818.65032)] that checking stability of symmetric interval matrices is NP-hard, see also [\textit{A. Nemirovskii}, ``Several NP-hard problems arising in robust stability analysis'', Math. Control Signals Syst. 6, 99--105 (1993; Zbl 0792.93100)]. However, this does not imply that polynomial-time algorithms cannot be devised for special classes of interval matrices, or that useful sufficient conditions for stability cannot be obtained. An example of a simple sufficient condition for stability of interval Jacobi matrices is given in Theorem 2 of the paper under review.