Stability and convex hulls of matrix powers

Invertibility of all convex combinations of A and I is equivalent to the real eigenvalues of A, if any, being positive. Invertibility of all matrices whose rows are convex combinations of the respective rows of A and I is equivalent to A having positive principal minors (i.e. being a P-matrix). Thes...

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Bibliographic Details
Published in:Linear & multilinear algebra 2018-04, Vol.66 (4), p.769-775
Main Authors: Torres, Patrick K., Tsatsomeros, Michael J.
Format: Article
Language:English
Subjects:
Convexity
Eigenvalues
Equivalence
Hulls
matrix hull
matrix powers
nonsingularity
P-matrix
positive stability
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Summary:Invertibility of all convex combinations of A and I is equivalent to the real eigenvalues of A, if any, being positive. Invertibility of all matrices whose rows are convex combinations of the respective rows of A and I is equivalent to A having positive principal minors (i.e. being a P-matrix). These results are extended by considering convex combinations of higher powers of A and of their rows. The invertibility of matrices in these convex hulls is associated with the eigenvalues of A lying in open sectors of the right-half plane and provides a general context for the theory of matrices with P-matrix powers.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2017.1322033