On Lie algebras, submanifolds and structured matrices

Abstract

I show that manifolds arise naturally when we consider structured eigen-problems for symmetric matrices. In particular, I show that the space of symmetric matrices is naturally partitioned into a collection S of connected submanifolds with the following property: For every symmetric matrix A, the submanifold in S containing A consists of matrices which have the same eigenvalues as A and the same staircase structure as A. I also show that the space of symmetric matrices is naturally partitioned into a collection T of connected submanifolds with the following property: For every symmetric matrix A, the submanifold in T containing A consists of matrices which have the same eigenvalues as A and the same displacement inertia as A. I obtain these results by considering appropriate Lie algebras of vector fields on the space of symmetric matrices.