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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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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description 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.
doi_str_mv 10.1080/03081087.2017.1322033
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source Taylor and Francis:Jisc Collections:Taylor and Francis Read and Publish Agreement 2024-2025:Science and Technology Collection (Reading list)
subjects Convexity
Eigenvalues
Equivalence
Hulls
matrix hull
matrix powers
nonsingularity
P-matrix
positive stability
title Stability and convex hulls of matrix powers
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