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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An application of theorems of Schur and Albert

Author: Thomas L. Markham
Journal: Proc. Amer. Math. Soc. 59 (1976), 205-210
MSC: Primary 15A48; Secondary 15A57
MathSciNet review: 0432682
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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 $P$.

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Keywords: Positive semidefinite, positive definite, Schur complement, <IMG WIDTH="27" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img12.gif" ALT="$M$">-matrices, class <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$P$">
Article copyright: © Copyright 1976 American Mathematical Society