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The main result is that a square matrix D is convergent (limn→∞Dn=0) if andonly if it is the Cayley transform CA=(I−A)−1(I+A) of a stable matrix A, where a stable matrixis one whose characteristic values all have negative real parts. In passing, the concept ofCayley transform is generalized, and the generalized version is shown closely related to theequation AG+GB=D. This gives rise to a characterization of the non-singularity of themapping X→AX+XB. As consequences are derived several characterizations of stability(closely related to Lyapunov's result) which involve Cayley transforms

Stable matrices, the Cayley transform, and convergent matrices