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∈Rn×n nonsingular, S being any fixed subspace of Rn×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.