# Orthogonality to matrix subspaces, and a distance formula

## TL;DR

This paper provides a precise condition for matrix orthogonality to subspaces and derives a formula for the distance from matrices to certain subalgebras, advancing understanding of matrix geometry.

## Contribution

It introduces a necessary and sufficient condition for matrix orthogonality to subspaces and a new distance formula to unital C*-subalgebras.

## Key findings

- Characterization of Birkhoff-James orthogonality to subspaces
- Explicit distance formula to unital C*-subalgebras
- Enhanced understanding of matrix subspace geometry

## Abstract

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