Free Semidefinite Representation of Matrix Power Functions

TL;DR

This paper demonstrates that the hypograph and epigraph of matrix power functions have free semidefinite representations when the exponent is rational, extending the understanding of convexity and representation of these functions.

Contribution

It provides the first known free semidefinite representations for the hypograph and epigraph of matrix power functions with rational exponents.

Findings

Hypograph of X^p has free semidefinite representation for p in [0,1] when rational.

Epigraph of X^p has free semidefinite representation for p in [-1,0] or [1,2] when rational.

Results extend semidefinite representability to a broader class of matrix power functions.

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.