Stability Analysis, Convex Hulls of Matrix Powers and their Relations to P-matrices
Torres, Patrick Kisa
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Invertibility of all convex combinations of an $n\times n$ matrix $A$ and the $n\times n$ identity matrix $I$ is equivalent to the real eigenvalues of $A$, if any, being positive. Moreover, 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). In this dissertation, we extend these results 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 open right half-plane and provides a general context for the theory of matrices with P-matrix powers. We present a new result regarding the spectrum of $A$ that holds as a consequence of assuming the invertibility of all infinite convex combinations of $A$ and its powers.