A real or complex square matrix \(A\) is said to be Schur stable if \(\rho(A)\) (spectral radius of \(A\)) \(< 1\), Schur \(D\)-stable if \(\rho(AD) < 1\) for every real diagonal matrix \(D\) with \(|D|\) (entries of \(D\)) \(\leq I\), where \(I\) is the identity matrix, and vertex stable if \(\rho(AD) < 1\) for every real diagonal matrix \(D\) with \(|D|= I\). It is proved that a real \(3 \times 3\) matrix \(A\) is Schur \(D\)-stable if and only if \(A\) is vertex stable. It is also shown for principally nilpotent \(n \times n\) complex matrices that they are perfectly Schur \(D\)-stable i.e. \(\lambda A\) belongs to the set of all \(n \times n\) complex Schur \(D\)-stable matrices for all eigenvalues \(\lambda \epsilon {\mathbb C}\). Properties of diagonally stable and simultaneously diagonally stable matrices are also discussed.

Reviewer: Vaclav Burjan (Praha)

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