Matrix manifolds and the Jordan structure of the bialternate matrix product

Abstract

The bialternate product of matrices was introduced at the end of the 19th century and recently revived as a computational tool in problems where real matrices with conjugate pairs of pure imaginary eigenvalues are important, i.e., in stability theory and Hopf bifurcation problems. We give a complete description of the Jordan structure of the bialternate product 2A⊙I n of an n×n matrix A, thus extending several partial results in the literature. We use these results to obtain regular (local) defining systems for some manifolds of matrices which occur naturally in applications, in particular for manifolds with resonant conjugate pairs of pure imaginary eigenvalues. Such defining systems can be used analytically to obtain local parameterizations of the manifolds or numerically to set up Newton systems with local quadratic convergence. We give references to explicit numerical applications and implementations in software. We expect that the analysis provided in this paper can be used to further improve such implementations.