On calculating generalized Cayley transforms of bounded norm

Abstract

Given an n×n orthogonal matrix Q, there exists a diagonal matrix D with each diagonal entry chosen from { − 1,1}, such that QD + I is non-singular and such that if

$$ S=(QD-I)(QD+I)^{-1}, $$

then the skew matrix S has every element in the interval [ − 1,1]. We prove that such a D exists and show that it can be computed efficiently and reliably.