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Perron-Frobenius theorem (Theorem)

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 $ \vert\rho(A)\vert\geq \vert\lambda\vert$, 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)>\vert\lambda\vert$ for all $ \lambda\in\sigma(A)$, $ \lambda\neq\rho(A)$.



"Perron-Frobenius theorem" is owned by jarino.
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See Also: fundamental theorem of demography
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Cross-references: primitive matrix, positive, simple, irreducible matrix, addition, eigenvalue, spectral radius, spectrum, matrix
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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 8310 times total.

Classification:
AMS MSC: 15A18 (Linear and multilinear algebra; matrix theory :: Eigenvalues, singular values, and eigenvectors)
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