Smith Forms of Palindromic Matrix Polynomials

Many applications give rise to matrix polynomials whose coefficients have a kind of reversal symmetry, a structure we call palindromic. Several properties of scalar palin-dromic polynomials are derived, and together with properties of compound matrices, used to establish the Smith form of regular and singular T-palindromic matrix polyno-mials over arbitrary fields. The invariant polynomials are shown to inherit palindromic-ity, and their structure is described in detail. Jordan structures of palindromic matrix polynomials are characterized, and necessary conditions for the existence of structured linearizations established. In the odd degree case, a constructive procedure for building palindromic linearizations shows that the necessary conditions are sufficient as well. The Smith form for ∗-palindromic polynomials is also analyzed. Finally, results for palindromic matrix polynomials over fields of characteristic two are presented