*(English)*Zbl 0547.15009

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

The diagonally stable matrices $\mathcal{A}$ 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 $\mathcal{L}$ consisting of those matrices A such that $AX+X{A}^{T}$ is positive definite for some positive definite matrix X, the P-matrices $\mathcal{P}$ consisting of those A all of whose principal minors are positive, and the semi-positive matrices $\mathcal{S}$ consisting of all A such that $Ax>0$ for some $x>0\xb7$

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 $n\times n$ matrix A is the graph whose vertices are 1,2,..., and whose edges are the pairs $\{$ i,$j\}$, $i\ne j$, 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 $\mathcal{L}\Leftarrow \mathcal{A}\iff \mathcal{P}\Rightarrow \mathcal{S}$ where $\Rightarrow $ 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.

##### MSC:

15A48 | Positive matrices and their generalizations (MSC2000) |

15A18 | Eigenvalues, singular values, and eigenvectors |

15A42 | Inequalities involving eigenvalues and eigenvectors |