Normal form of m-by-n-by-2 matrices for equivalence

Abstract

We study m × n × 2 matrices up to equivalence and give a canonical form of m × 2 × 2 matrices over any field.

Congruence of matrix spaces, matrix tuples, and multilinear maps

“…Some classes of matrix 2-tuples are classified up to weak equivalence in [2,3,9]. By [4,5], the problem of classifying matrix 3-tuples up to weak equivalence is wild, and so it contains the problems of classifying each system of linear maps and representations of each finite dimensional algebra; see [6] and [1, Proposition 9.14].…”

Wildness of the problems of classifying two-dimensional spaces of commuting linear operators and certain Lie algebras

“…with nonsingular S P F nˆn (the spaces V and S ´1V S are matrix isomorphic; see [14]). In Theorem 1(a), we prove the wildness of the problem of classifying two-dimensional vector spaces V Ă F nˆn of commuting matrices up to transformations (2). Each two-dimensional vector space V Ă F nˆn is given by its basis A, B P V that is determined up to transformations pA, Bq Þ Ñ pαA `βB, γA `δBq, in which " α γ β δ ‰ P F 2ˆ2 is a change-of-basis matrix.…”

“…Each two-dimensional vector space V Ă F nˆn is given by its basis A, B P V that is determined up to transformations pA, Bq Þ Ñ pαA `βB, γA `δBq, in which " α γ β δ ‰ P F 2ˆ2 is a change-of-basis matrix. Thus, the problem of classifying two-dimensional vector spaces V Ă F nˆn up to transformations (2) is the problem of classifying pairs of linear independent matrices A, B P F nˆn up to transformations pA, Bq Þ Ñ pA 1 , B 1 q :" S ´1pαA `βB, γA `δBqS,…”

“…The analogous problem of classifying matrix pairs pA, Bq up to weak congruence S T pαA `βB, γA `δBqS appears in the problem of classifying finite p-groups of nilpotency class 2 with commutator subgroup of type pp, pq, in the problem of classifying commutative associative algebras with zero cube radical, and in the problem of classifying Lie algebras with central commutator subalgebra; see [3,4,6,18]. The problem of classifying matrix pairs up to weak equivalence RpαA `βB, γA `δBqS appears in the theory of tensors [2].…”

On the Triangular Form of a Polynomial Matrix and Its Invariants with Respect to the Semiscalar Equivalence

“…This problem was studied in numerous works (see, e.g., [1,2,12]). Many classification problems, in particular, the problem of equivalence of spatial matrices [10], can be reduced to the problem mentioned above.…”