Orthogonality to matrix subspaces, and a distance formula

Abstract

We obtain a necessary and sufficient condition for a matrix $A$ to be Birkhoff-James orthogonal to any subspace $\mathcal{W}$ of ${\mathbb{M}}_n(\mathbb{C})$. Using this we obtain an expression for the distance of $A$ from any unital $C^{\ast }$ subalgebra of ${\mathbb{M}}_n(\mathbb{C})$.

12 pages