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Perron-Frobenius theorem
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(Theorem)
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Let $A$ be a nonnegative matrix. Denote its spectrum by $\sigma(A)$ Then the spectral radius $\rho(A)$ is an eigenvalue, that is, $\rho(A)\in \sigma(A)$ and is associated to a nonnegative eigenvector.
If, in addition, $A$ is an irreducible matrix, then $|\rho(A)|\geq |\lambda|$ for all $\lambda\in \sigma(A)$ $\lambda\neq \rho(A)$ and $\rho(A)$ is a simple eigenvalue associated to a positive eigenvector.
If, in addition, $A$ is a primitive matrix, then $\rho(A)>|\lambda|$ for all $\lambda\in\sigma(A)$ $\lambda\neq\rho(A)$
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"Perron-Frobenius theorem" is owned by jarino.
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Cross-references: primitive matrix, positive, simple, irreducible matrix, addition, eigenvalue, spectral radius, spectrum, matrix
There is 1 reference to this entry.
This is version 2 of Perron-Frobenius theorem, born on 2002-12-22, modified 2003-12-01.
Object id is 3812, canonical name is PerronFrobeniusTheorem.
Accessed 9508 times total.
Classification:
| AMS MSC: | 15A18 (Linear and multilinear algebra; matrix theory :: Eigenvalues, singular values, and eigenvectors) |
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Pending Errata and Addenda
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