International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 1, Pages 77-81

Stable matrices, the Cayley transform, and convergent matrices

Tyler Haynes

Mathematics Department, Saginaw Valley State University, University Center 48710, Michigan, USA

Received 17 January 1990

Copyright © 1991 Tyler Haynes. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The main result is that a square matrix D is convergent (limnDn=0) if and only if it is the Cayley transform CA=(IA)1(I+A) of a stable matrix A, where a stable matrix is one whose characteristic values all have negative real parts. In passing, the concept of Cayley transform is generalized, and the generalized version is shown closely related to the equation AG+GB=D. This gives rise to a characterization of the non-singularity of the mapping XAX+XB. As consequences are derived several characterizations of stability (closely related to Lyapunov's result) which involve Cayley transforms.