[00770] Rellich eigendecomposition of paraHermitian matrices, with applications

Session Time & Room : 1E (Aug.21, 17:40-19:20) @G304

Type : Contributed Talk

Abstract : Let $H(z)$ be paraHermitian, that is, analytic and Hermitian on the unit circle $S^1$. We prove that $H(z)=U(z)D(z)U(z)^P$ where, for all $z \in S^1$, $U(z)$ is unitary, $U(z)^P=U(z)^*$, and $D(z)$ is real diagonal; moreover, $U(z), D(z)$ are analytic in $w=z^{1/N}$ for some positive integer $N$, and $U(z)^P$ is the paraHermitian conjugate of $U(z)$. We discuss the implications on the svd of an $S^1$-analytic matrix and the sign characteristics of unimodular eigenvalues of $*$-palindromic matrix polynomials.