Shi, Zhicheng; Gao, Weibin A necessary and sufficient condition for the positive-definiteness of interval symmetric matrices. (English) Zbl 0613.65042 Int. J. Control 43, 325-328 (1986). Let \(P=(p_{ij})\in {\mathbb{R}}^{n\times n}\) and \(Q=(q_{ij})\in {\mathbb{R}}^{n\times n}\) be symmetric matrices with \(p_{ij}\leq q_{ij}\) for \(i,j=1,...,n\). Define \(S(P,Q):=\{A| A=(a_{ij})\in {\mathbb{R}}^{n\times n}\), \(A\) symmetric, \(p_{ij}\leq a_{ij}\leq q_{ij}\) for \(i,j=1,...,n\}\) and \(V(P,Q):=\{A| A=(a_{ij})\in {\mathbb{R}}^{n\times n}\), \(A\) symmetric, \(p_{ij}=a_{ij}\) or \(a_{ij}=q_{ij}\) for \(i,j=1,...,n\) with \(i\neq j\), \(a_{ii}=p_{ii}\) for \(i=1,...,n\}\). Then V(P,Q) is a finite subset of S(P,Q). Geometrically speaking, any element of V(P,Q) is a vertex of S(P,Q). The authors show: If the finitely many matrices in V(P,Q) are positive- definite, then all matrices in S(P,Q) are positive-definite and \(\min \{\det (A)| A\in V(P,Q)\}=\min \{\det (A)| A\in S(P,Q)\}\). This result may serve as a criterion in testing positive-definiteness of bounded time-dependent symmetric real matrices. Reviewer: H.Fischer Cited in 19 Documents MSC: 65F30 Other matrix algorithms (MSC2010) 65G30 Interval and finite arithmetic 65F40 Numerical computation of determinants 15A42 Inequalities involving eigenvalues and eigenvectors 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory Keywords:interval analysis; stability; control theory; time-dependent symmetric real matrices PDFBibTeX XMLCite \textit{Z. Shi} and \textit{W. Gao}, Int. J. Control 43, 325--328 (1986; Zbl 0613.65042) Full Text: DOI References: [1] DOI: 10.1080/00207178408933235 · Zbl 0535.93055 [2] DOI: 10.1080/00207178308933004 · Zbl 0518.93052 [3] HE J. -X., Ordinary Differential Equations (1981) [4] DOI: 10.1080/00207178408933248 · Zbl 0539.65017 [5] DOI: 10.1080/00207178408933211 · Zbl 0535.93056 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.