Stable matrices, the Cayley transform, and convergent matrices

Abstract

The main result is that a square matrix D is convergent (limn→∞Dn=0) if and only if it is the Cayley transform CA=(I−A)−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 X→AX+XB. As consequences are derived several characterizations of stability (closely related to Lyapunov's result) which involve Cayley transforms.

Keywords

• stable matrix
• Cayley transform
• convergent matrix.