An extension of Birkhoff's theorem with an application to determinants

摘要

Let M, (F) denote the algebra of n x n matrices over the field F of complex, or real, numbers. Given a self-adjoint involution J epsilon M-n (C), that is, J = J*, J(2) = I, let us consider U endowed with the indefinite inner product [.] induced by J and defined by [x, y] := (Jx, y), x,y epsilon C-n. Assuming that (r, n - r), 0 <= r <= n, is the inertia of J, without loss of generality we may assume J = diag(j(1),..., j,,) For T = (vertical bar tik vertical bar(2)) epsilon M-n(R), the matrices of the form T = (vertical bar tik vertical bar(2) 12.j(i) j(k)), With all line sums equal to 1, are called J-doubly stochastic matrices. In the particular case r E {0, n}, these matrices reduce to doubly stochastic matrices, that is, non-negative real matrices with all line sums equal to 1. A generalization of Birkhoff's theorem on doubly stochastic matrices is obtained for J-doubly stochastic matrices and an application to determinants is presented. (c) 2007 Elsevier Inc. All rights reserved.