Positive definiteness of Hermitian interval matrices (Q846327)

The positive definiteness of Hermitian interval matrices is investigated. It is shown that for each Hermitian matrix \(A\) in a Hermitian interval matrix \(A\) there exists a Hermitian vertex matrix \(B\in A\) such that \(\Lambda_n (A)\geq \Lambda_n (B)\), where \(\Lambda_n\) denotes the minimal eigenvalue. This implies that positive definiteness of all Hermitian vertex matrices in \(A\) implies positive definiteness of each Hermitian matrix in \(A\), which is a generalization of a similar result for symmetric interval matrices.