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arXiv · 1305.4289

Free Semidefinite Representation of Matrix Power Functions

J. William Helton·Jiawang Nie·Jeremy S. Semko
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Abstract

Consider the matrix power function X^p defined over the cone of positive definite matrices S^{n}_{++}. It is known that X^p is convex over S^{n}_{++} if p is in [-1,0] or [1,2] and X^p is concave over S^{n}_{++} if p is in [0,1]. We show that the hypograph of X^p admits a free semidefinite representation if p in [0,1] is rational, and the epigraph of X^p admits a free semidefinite representation if p in [-1,0] or [1,2] is rational.

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J. William Helton, Jiawang Nie, Jeremy S. Semko. 2014-10-09. Free Semidefinite Representation of Matrix Power Functions. https://arxiv.org/abs/1305.4289

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