Complex orthogonal and symplectic matrices depending on parameters

Abstract

Versal deformations of elements of complex orthogonal and symplectic Lie algebras are studied. For a general element M of o(n,C) or sp(2n,C), a normal form MA is found which, unlike the Jordan normal form MJ, depends holomorphically on M and on the similarity transformation MA = gMg−1 from the corresponding group. Orthogonal and symplectic cases are treated simultaneously in order to underline their close relation. Bundles of matrices of low codimension are listed and bifurcation diagrams of two-parameter families of orthogonal matrices are shown. Finally, versal deformations of all elements of o(6,C) are explicitly shown.