International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 1, Pages 77-81
Stable matrices, the Cayley transform, and convergent matrices
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 is convergent () if and
only if it is the Cayley transform of a stable matrix , 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 . This gives rise to a characterization of the non-singularity of the
mapping . As consequences are derived several characterizations of stability
(closely related to Lyapunov's result) which involve Cayley transforms.