An invariant for a subspace of the finite dimensional vector space and automorphism partition of a real symmetric matrix (Q1103019)

Given an m-dimensional subspace of an n-dimensional linear space, the authors consider the set of permutation matrices P of order n for which there exist invertible matrices L of order m such that \(U=LUP\), where U is an m by n matrix formed from the vectors of a basis of the given subspace. It turns out that this definition is independent of the particular basis, and the corresponding set of permutation matrices is a group with respect to the usual multiplication of matrices. This is the automorphism group of the given subspace. The automorphism group of a given square matrix M consists of the permutation matrices P such that \(M=P^{-1}MP\). The two concepts are related via a symmetric matrix which represents a projection onto the given subspace. It is also shown that, given a real symmetric matrix, its automorphisms can be expressed in terms of the automorphisms of all of its eigenspaces.