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arXiv · 1712.08729

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

V. A. Bovdi·T. G. Gerasimova·M. A. Salim·V. V. Sergeichuk
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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.

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V. A. Bovdi, T. G. Gerasimova, M. A. Salim, V. V. Sergeichuk. 2017-12-23. Reduction of a pair of skew-symmetric matrices to its canonical form under congruence. https://doi.org/10.1016/j.laa.2017.12.013

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