Stable matrices, the Cayley transform, and convergent matrices
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...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1991-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171291000078 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |