The sum of orthogonal matrices

Abstract

Let F∈{R,C,H}. Let Un(F) be the set of unitary matrices in Mn(F), and let On(F) be the set of orthogonal matrices in Mn(F). Suppose n⩾2. We show that every A∈Mn(F) can be written as a sum of matrices in Un(F) and of matrices in On(F). Let A∈Mn(F) be given and let k⩾2 be the least integer that is a least upper bound of the singular values of A. When F=C, we show that A can be written as a sum of k matrices from Un(F). When F=R, we show that if k⩽3, then A can be written as a sum of 6 orthogonal matrices; if k⩾4, we show that A can be written as a sum of k+2 orthogonal matrices.