Real Polynomial Gram Matrices Without Real Spectral Factors

Puoya Tabaghi; Ivan Dokmanić

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arXiv · 1903.04085

Real Polynomial Gram Matrices Without Real Spectral Factors

Puoya Tabaghi·Ivan Dokmanić

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Abstract

It is well known that a non-negative definite polynomial matrix (a polynomial Gramian) $G(t)$ can be written as a product of its polynomial spectral factors, $G(t) = X(t)^H X(t)$. In this paper, we give a new algebraic characterization of spectral factors when $G(t)$ is real-valued. The key idea is to construct a representation set that is in bijection with the set of real polynomial Gramians. We use the derived characterization to identify the set of all complex polynomial matrices that generate real-valued Gramians, and we formulate a conjecture that typical rank-deficient real polynomial Gramians have real spectral factors.

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Puoya Tabaghi, Ivan Dokmanić. 2019-03-11. Real Polynomial Gram Matrices Without Real Spectral Factors. https://arxiv.org/abs/1903.04085

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