An application of theorems of Schur and Albert

Abstract:

Suppose ${\Pi _n}$ is the cone of $n \times n$ positive semidefinite matrices, and $\operatorname {int} ({\Pi _n})$ is the set of positive definite matrices. Theorems of Schur and Albert are applied to obtain some elements of ${\Pi _n}$ and $\operatorname {int} ({\Pi _n})$. Then an analogue of Albert’s theorem is given for $M$-matrices, and finally a generalization is given for matrices of class $P$.