Canonical forms of pairs of complex matrices (Q753893)

Representation theory of finite-dimensional associative algebras is used in order to classify explicitely the pairs of real subspaces of a finite- dimensional complex vector space in terms of complex matrices. In this respect one classifies the orbits of the direct product \(GL(m,{\mathbb{C}})\times GL(m_ 1,{\mathbb{R}})\times GL(m_ 2,{\mathbb{R}})\) acting on the set of pairs of \(m\times m_ i\) complex matrices \(A_ i\), \(i=1,2\), by \((A_ 1,A_ 2)(Q,P_ 1,P_ 2)=(QA_ 1P_ 1^{-1},QA_ 2P_ 2^{-1}).\) As a special case one resolves the general case of canonical forms of pairs of complex matrices.