Normal forms in matrix spaces
The action of the group \(G_{lr}[m,n] \equiv GL(m,\mathbb{C}) \times GL(n,\mathbb{C})\) on the space of \(m \times n\) matrices over \(\mathbb{C}\) is classified and certain class of subgroups of \(G_{lr}[m,n]\) which are called admissible are introduced. An algorithm that reduces any matrix to a normal form with respect to the action of an admissible group is given. The algorithm is based on the fact that the action of an admissible group corresponds to a partition of an \(m \times n\) matrix into blocks, each block corresponding to a pair consisting of diagonal block of \(P \in GL(m,\mathbb{C})\) and a diagonal block of \(Q \in GL(n,\mathbb{C})\). The algorithm works inductively, at each step it chooses one block, transforms it to a normal form and replaces the group by the stabilizer of this normal form. The process ends in a finite number of steps, producing the canonical form. A difficulty of this approach is that the stationary subgroups of the usual normal forms with respect to the actions of \(G_{lr}[m,n]\) are not admissible and this is an obstruction for inductive steps of the algorithm. To overcome this difficulty the other normal forms, which are called modified, are introduced. This modification is useful also for other similar problems as e. g. for the reduction to a normal form of quaternion matrices by unitary similarity.
scientific article; zbMATH DE number 1556614