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Matrix diagonal stability and its implications. (English) Zbl 0547.15009

This paper considers inclusion relations between the following four classes of real square matrices:

The diagonally stable matrices $𝒜$ consisting of those matrices A such that $AD+D{A}^{T}$ is positive definite for some positive definite diagonal matrix D, the positive stable matrices $ℒ$ consisting of those matrices A such that $AX+X{A}^{T}$ is positive definite for some positive definite matrix X, the P-matrices $𝒫$ consisting of those A all of whose principal minors are positive, and the semi-positive matrices $𝒮$ consisting of all A such that $Ax>0$ for some $x>0·$

In particular, it is shown that if A is a P-matrix and if the nondirected graph of A is a forest, then A is diagonally stable. (The nondirected graph of an $n×n$ matrix A is the graph whose vertices are 1,2,..., and whose edges are the pairs $\left\{$ i,$j\right\}$, $i\ne j$, for which either ${a}_{ij}$ or ${a}_{ji}$ is nonzero.) This result, in view of previously known results, yields for those matrices whose nondirected graph is a forest the relations $ℒ⇐𝒜⇔𝒫⇒𝒮$ where $⇒$ means ”is contained in” and the absence of an implication implies a counterexample. Partial results and some open questions are given for real spectra matrices and for a class of matrices called $\omega$ -matrices.

Reviewer: J.Brawley
##### MSC:
 15A48 Positive matrices and their generalizations (MSC2000) 15A18 Eigenvalues, singular values, and eigenvectors 15A42 Inequalities involving eigenvalues and eigenvectors