Free Semidefinite Representation of Matrix Power Functions
Abstract
Consider the matrix power function Xp defined over the cone of positive definite matrices Sn++. It is known that Xp is convex over Sn++ if p is in [-1,0] or [1,2] and Xp is concave over Sn++ if p is in [0,1]. We show that the hypograph of Xp admits a free semidefinite representation if p in [0,1] is rational, and the epigraph of Xp admits a free semidefinite representation if p in [-1,0] or [1,2] is rational.
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