A min–max theorem for complex symmetric matrices

Abstract

We optimize the form Re x t Tx to obtain the singular values of a complex symmetric matrix T. We prove that for 0 ⩽ k < n 2 , min codim V = k max x ∈ V ‖ x ‖ = 1 Re x t Tx = σ 2 k + 1 , where T is an n × n complex symmetric matrix having singular values σ 1 ⩾ ⋯ ⩾ σ n . We also show that the singular values missed in this theorem (i.e. σ 2, σ 4, …) are obtained by a similar optimization over real subspaces.