Research Seminar in Mathematics - On the consistency of the matrix equation X^TAX=B when B is either symmetric or skew

Speaker

Fernando De Terán, Universidad Carlos III de Madrid.

Abstract

In this talk, we analyze the consistency of the matrix equation 

X^T AX=B,           (1)

where A in an mxm complex matrix, X in an mxn complex matrix (unknown), and B (complex nxn) is either symmetric or skew-symmetric (and (·)^T means the transpose). In particular, we first provide a necessary condition for (1) to have a solution X. Then, we will prove that this condition is also sufficient for most matrices A and an arbitrary symmetric (or skew) matrix B.

We want to emphasize that the question on the consistency of (1), when B is symmetric (respectively, skew), is equivalent to the following problem: given a bilinear form over C^m (represented by the matrix A), find the maximum dimension of a subspace such that the restriction of the bilinear form to this subspace is a symmetric (resp., skew) non-degenerate bilinear form.

Welcome!