Reduction of a pair of skew-symmetric matrices to its canonical form under congruence

Abstract

Let $(A,B)$ be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum \[ (\underline{\underline A},\underline{\underline B})\oplus (A_1,B_1)\oplus\dots\oplus(A_t,B_t) \] that is congruent to $(A,B)$, in which $(\underline{\underline{A}},\underline{\underline{B}})$ is a pair of nonsingular matrices and $(A_1,B_1),$ $\dots ,$ $(A_t,B_t)$ are singular indecomposable canonical pairs of skew-symmetric matrices under congruence. We give an algorithm that constructs a regularization decomposition. We also give a constructive proof of the known canonical form of $(A,B)$ under congruence over an algebraically closed field of characteristic not 2.

16 pages

References in corpus (3)

• Classification problems for system of forms and linear mappings
• A regularization algorithm for matrices of bilinear and sesquilinear forms
• Computation of the canonical form for the matrices of chains and cycles of linear mappings