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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A substitute of l'Hospital's rule for matrices


Author: W. Kratz
Journal: Proc. Amer. Math. Soc. 99 (1987), 395-402
MSC: Primary 15A24; Secondary 15A45, 34B25, 34C11, 58F05
MathSciNet review: 875370
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Abstract: In this paper the following limit theorem is obtained: If $ A$ and $ B$ are $ (n,n)$-matrices with $ {\text{rank}}({A^T},{B^T}) = n,\;{A^T}B = {B^T}A$, then $ A{(A + SB)^{ - 1}}S \to 0$ as $ S \to 0 + ,\;{\text{i}}{\text{.e}}{\text{.}}\;{\text{S}} \to {\text{0}}$ where $ S$ is symmetric and positive definite. Some applications of this result are given to linear algebra (the behavior of $ {(A + \lambda B)^{ - 1}}$ as $ \lambda \to 0)$ and to differential equations (the asymptotic behavior of Hamiltonian systems and of selfadjoint differential equations of even order).


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0875370-4
PII: S 0002-9939(1987)0875370-4
Keywords: Hamiltonian system, Riccati equation, symmetric matrices, matrix functions, Picone identity, focal points
Article copyright: © Copyright 1987 American Mathematical Society