Claremont Colleges Analysis Seminar
2008-2009

Title: Unitary equivalence to a complex symmetric matrix

Abstract:   It is well-known that any $n \times n$ complex matrix is \emph{similar} to a complex symmetric matrix (CSM) (i.e., $A = A^t$).  Indeed, as opposed to a selfadjoint matrix (i.e., $A = A^*$), a CSM can have any possible Jordan form.  This makes the problem of determining exactly which matrices / operators are \emph{unitarily equivalent} to a complex symmetric matrix (UECSM) somewhat difficult. In particular, we cannot employ any of the standard similarity invariants from linear algebra. We discuss several recent solutions to this problem.  To this end, we also discuss the basic structure of \emph{complex symmetric operators} on Hilbert space, highlighting some surprising members of this family, and a few applications.