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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| Format: | Article |
| Language: | English |
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Wiley
1991-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171291000078 |
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| _version_ | 1832564352565116928 |
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| author | Tyler Haynes |
| author_facet | Tyler Haynes |
| author_sort | Tyler Haynes |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-b132f32e68544d2e9fb8b73dd1fb910d |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1991-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-b132f32e68544d2e9fb8b73dd1fb910d2025-02-03T01:11:16ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251991-01-01141778110.1155/S0161171291000078Stable matrices, the Cayley transform, and convergent matricesTyler Haynes0Mathematics Department, Saginaw Valley State University, University Center 48710, Michigan, USAThe 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.http://dx.doi.org/10.1155/S0161171291000078stable matrixCayley transformconvergent matrix. |
| spellingShingle | Tyler Haynes Stable matrices, the Cayley transform, and convergent matrices International Journal of Mathematics and Mathematical Sciences stable matrix Cayley transform convergent matrix. |
| title | Stable matrices, the Cayley transform, and convergent matrices |
| title_full | Stable matrices, the Cayley transform, and convergent matrices |
| title_fullStr | Stable matrices, the Cayley transform, and convergent matrices |
| title_full_unstemmed | Stable matrices, the Cayley transform, and convergent matrices |
| title_short | Stable matrices, the Cayley transform, and convergent matrices |
| title_sort | stable matrices the cayley transform and convergent matrices |
| topic | stable matrix Cayley transform convergent matrix. |
| url | http://dx.doi.org/10.1155/S0161171291000078 |
| work_keys_str_mv | AT tylerhaynes stablematricesthecayleytransformandconvergentmatrices |