A canonical form for palindromic pencils and palindromic factorizations
Christian Schröder · 2006
We consider real or complex palindromic pencils, i.e., pencils of the form $A-\lambda A^\star$, where $A^{\star}$ denotes either the transpose $A^T$ or the conjugate transpose $A^*$. Structured canonical forms for these pencils are derived that reveal complete spectral information. Moreover, this canonical form is used to provide necessary and sufficient conditions for the existence of palindromic factorizations $B=A^{-\star}A$ of a given square matrix $B$. In particular, we answer the questions when symplectic matrices allow palindromic factorization and when a matrix having a palindromic factorization is similar to a symplectic one.