Orthogonality of matrices

Let X be a real finite-dimensional normed space with unit sphere SX and let L(X)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∈L(X)A⊥C⇔∃u∈SX:‖A‖=‖Au‖,Au⊥Cu,where ⊥ denotes Birkhoff orthogonality.