Standard triples of structured matrix polynomials

The notion of standard triples plays a central role in the theory of matrix polynomials. We study such triples for matrix polynomials P(λ) with structure S, where S is the Hermitian, symmetric, ★-even, ★-odd, ★-palindromic or ★-antipalindromic structure (with ★=∗,T). We introduce the notion of 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(λ) has structure S if and only if P(λ) admits an S-structured standard triple, and moreover that every standard triple of a matrix polynomial with structure S is S-structured. We investigate the important special case of S-structured Jordan triples.