Gram-like matrix preserving extensions and completions of noncommutative polynomials

Arijit Mukherjee; Arindam Sutradhar

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arXiv · 2501.12063

Gram-like matrix preserving extensions and completions of noncommutative polynomials

Arijit Mukherjee·Arindam Sutradhar

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Abstract

Given a positive noncommutative polynomial $f$, equivalently a sum of Hermitian squares (SOHS), there exists a positive semidefinite Gram matrix that encrypts all the structural essence of $f$. There are no available methods for extending a noncommutative polynomial to a SOHS keeping the Gram matrices unperturbed. As a remedy, we introduce an equally significant notion of Gram-like matrices and provide linear algebraic techniques to get the desired extensions. We further use positive semidefinite completion problem to get SOHS and provide criteria in terms of chordal graphs and 2-regular projective algebraic sets.

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BibTeX RIS

Arijit Mukherjee, Arindam Sutradhar. 2025-01-21. Gram-like matrix preserving extensions and completions of noncommutative polynomials. https://arxiv.org/abs/2501.12063

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