The bialternate product of matrices was introduced at the end of the 19th c
entury 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 x n matrix A, thus extending several partial results in the literatur
e. 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 parameteriz
ations of the manifolds or numerically to set up Newton systems with local
quadratic convergence. We give references to explicit numerical application
s and implementations in software. We expect that the analysis provided in
this paper can be used to further improve such implementations. (C) 1999 El
sevier Science Inc. All rights reserved.