Symmetric matrices and congruence

This chapter develops diagonal canonical forms and proves inertia theorems for hermitian and skewhermitian matrices with respect to involutions (including the conjugation). These canonical forms enable the identification of maximal neutral and maximal semidefinite subspaces, with respect to a given hermitian or skewhermitian matrix, in terms of their dimensions, which the chapter also discusses. The chapter begins this discussion by introducing canonical forms under congruence, defined as two matrices A , B - ∈ H n×n , which are said to be congruent if A = S * BS for some invertible S - ∈ H n×n . If φ‎ is a nonstandard involution, then A,B - ∈ H n×n are φ‎-congruent if A = S ᵩ BS for some invertible S .