Orthogonality of matrices

Abstract

Let X be a real finite-dimensional normed space with unit sphere S X 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 ∈ S X : ‖ A ‖ = ‖ Au ‖ , Au ⊥ Cu , where ⊥ denotes Birkhoff orthogonality.