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Uncoupling the Perron eigenvector problem. (English) Zbl 0673.15006
A method is given to find the unique normalized Perron vector $$\pi$$ satisfying $$A\pi =\rho \pi$$ where A is a nonnegative irreducible $$m\times m$$ matrix with spectral radius $$\rho$$, $$\pi =(\pi_ 1,-\pi_ m)^ T$$ and $$\pi_ 1+...+\pi_ m=1$$. The matrix is uncoupled into two or more smaller matrices $$P_ 1,P_ 2,...,P_ k$$ such that this sequence has the following properties: (1) Each $$P_ i$$ is irreducible and nonnegative and has a unique Perron vector $$\pi^{(i)}$$. (2) Each $$P_ i$$ has the spectral radius $$\rho$$. (3) The Perron vectors $$\pi^{(i)}$$ for $$P_ i$$ can be determined independently. (4) The smaller Perron vectors $$\pi^{(i)}$$ can easily be coupled back together to form the Perron vector $$\pi$$ for A.
Reviewer: B.Ruffer-Beedgen

MSC:
 15B48 Positive matrices and their generalizations; cones of matrices 15A18 Eigenvalues, singular values, and eigenvectors
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References:
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