Real Factorization of Positive Semidefinite Matrix Polynomials
Sarah Gift, Hugo J. Woerdeman

TL;DR
This paper characterizes when a real symmetric positive semidefinite matrix polynomial can be factored into a product of a matrix polynomial and its transpose, providing a constructive proof and an explicit algorithm.
Contribution
It offers a necessary and sufficient condition for factorization based on the determinant and presents a constructive proof with an algorithm for computation.
Findings
Factorization exists if and only if the determinant is a perfect square.
Provides a constructive proof using a skew-symmetric solution to a Riccati equation.
Includes a detailed algorithm for computing the factorization.
Abstract
Suppose is a real regular symmetric positive semidefinite matrix polynomial. Then it can be factored as where is a real matrix polynomial with degree half that of if and only if is the square of a nonzero real polynomial. We provide a constructive proof of this fact, rooted in finding a skew-symmetric solution to a modified algebraic Riccati equation where are real matrices with and real symmetric. In addition, we provide a detailed algorithm for computing the factorization.
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Taxonomy
TopicsMatrix Theory and Algorithms · Tensor decomposition and applications · Advanced Optimization Algorithms Research
