ELA, Volume 9, pp. 190-196, September 2002, abstract.  

On the Cayley Transform of Positivity Classes 
of Matrices

Shaun M. Fallat and Michael J. Tsatsomeros

The Cayley transform of A, F=(I+A)^{-1}(I-A), is 
studied when A is a P-matrix, an M-matrix, an inverse 
M-matrix, a positive definite matrix, or a totally 
nonnegative matrix. Given a matrix A in each of these 
positivity classes and using the fact that the Cayley 
transform is an involution, properties of the ensuing 
factorization A=(I+F)^{-1}(I-F) are examined. 
Specifically, it is investigated whether these factors 
belong to the same positivity class as A and, conversely, 
under what conditions on these factors does A belong to 
one of the above positivity classes.