Geometrical and Spectral Properties of the Orthogonal Projections of the Identity

Abstract

We analyze the best approximation $AN$ (in the Frobenius sense) to the identity matrix in an arbitrary matrix subspace $AS$ ($A\in {ℝ}^{n\times n}$ nonsingular, $S$ being any fixed subspace of ${ℝ}^{n\times n}$). Some new geometrical and spectral properties of the orthogonal projection $AN$ are derived. In particular, new inequalities for the trace and for the eigenvalues of matrix $AN$ are presented for the special case that $AN$ is symmetric and positive definite.