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 sumis a pair of nonsingular matrices and (A 1 , B 1 ), . . . , (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.

Anticommutative Engel algebras of the first five levels

“…In the last case B ∼ = A by our arguments. Finally, T 4 (ǫ 5 23 ) → (1,5,6), (1,4,5),(1,1,3), (2,4,6),(2,3,5) T 3 (ǫ 5 34 ).…”

“…µ = T 2,2 (ǫ n−2 23 ). The classification of algebras of the form U ⋉ φ k 2 is strongly related to the classification of skew-symmetric matrix pairs considered, for example, in [1,4,16]. In fact, one has to factorize the classification obtained in these papers by an action of the group GL(k 2 ).…”

“…We have µ 1 15 = 0 by the nilpotence of the operator L e 5 . Replacing e i by µ 1 1,i−1 e 1 + e i for 3 i 5, we may assume that µ 1 1,i = 0 for all 2 i 5. Let us pick some α, β ∈ k and consider a new basis of V defined by the equalities f 1 = e 1 , f 2 = e 2 + αe 3 + βe 4 , f 3 = e 3 + αe 4 + βe 5 , e 4 = e 4 + αe 5 , f 5 = e 5 .…”