Minimum upper bound of the Cayley transform of an orthogonal matrix multiplied by signed permutation matrices

Abstract

O'Dorney has proved that for any orthogonal matrix U, there exists a signature matrix D (a diagonal matrix with diagonal entries ±1) by which |sij|≤1, where S=(sij)=Q(UD) is the Cayley transform of UD. In this paper, we prove that the upper bound can be reduced to 2−1 by multiplying by a certain signed permutation matrix, instead of a signature matrix.