Minimizing the Cayley transform of an orthogonal matrix by multiplying by signature matrices

Abstract

The Cayley transform, $(A)=(I−A)(I+A)−1, maps skew-symmetric matrices to orthogonal matrices and vice versa. Given an orthogonal matrix Q, we can choose a diagonal matrix D with each diagonal entry ±1 (a signature matrix) and, if I+QD is nonsingular, calculate the skew-symmetric matrix $(QD). An open problem is to show that, by a suitable choice of D, we can make every entry of $(QD) less than or equal to 1 in absolute value. We solve this problem by showing that the principal minors of $(QD) are related in a simple way to the principal minors of $(Q).