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## 2011.37: Standard Triples of Structured Matrix Polynomials

2011.37: Maha Al-Ammari and Francoise Tisseur (2011) Standard Triples of Structured Matrix Polynomials.

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## Abstract

The notion of standard triples plays a central role in the theory of matrix polynomials. We study such triples for matrix polynomials $P(\lambda)$ with structure $\mathcal{S}$, where $\mathcal{S}$ is the Hermitian, symmetric, $\star$-even, $\star$-odd, $\star$-palindromic or $\star$-antipalindromic structure (with $\star=*,T$). We introduce the notion of $\mathcal{S}$-structured standard triple. With the exception of $T$-(anti)palindromic matrix polynomials of even degree with both $-1$ and $1$ as eigenvalues, we show that $P(\lambda)$ has structure $\mathcal{S}$ if and only if $P(\lambda)$ admits an $\mathcal{S}$-structured standard triple, and moreover that every standard triple of a matrix polynomial with structure $\mathcal{S}$ is $\mathcal{S}$-structured. We investigate the important special case of $\mathcal{S}$-structured Jordan triples.

Item Type: MIMS Preprint MSC 2000 > 15 Linear and multilinear algebra; matrix theoryMSC 2000 > 65 Numerical analysis 2011.37 Dr Françoise Tisseur 10 May 2011