Orthogonality of matrices

Abstract

Let X be a real finite-dimensional normed space with unit sphere SX and let L (X) be the space of linear operators from X into itself. It is proved that X is an inner product space if and only if for A, C ∈ L (X)A ⊥ C ⇔ ∃ u ∈ SX : {norm of matrix} A {norm of matrix} = {norm of matrix} Au {norm of matrix}, Au ⊥ Cu,where ⊥ denotes Birkhoff orthogonality. © 2006 Elsevier Inc. All rights reserved.