Orthogonality to matrix subspaces, and a distance formula

We obtain a necessary and sufficient condition for a matrix $A$ to be Birkhoff-James orthogonal to any subspace $\mathcal{W}$ of $\mathbb M_n(\mathbb C)$\mathrm{.}Usingthisweobtainanexpressionforthedistanceof$A$fromanyunital$C^*$subalgebraof$\mathbb M_n(\mathbb C)$.