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Perron complement and Perron root. (English) Zbl 0999.15009
For a nonnegative irreducible matrix $$A$$ the well known Perron complement is generalized. The main theorem that describes the relation between the generalized Perron complement of $$A$$ and its Perron root, i.e. the spectral radius $$\rho(A)$$ of $$A$$, is stated. On several examples, it is shown how the use of the theorem can improve both lower and upper bounds for $$\rho(A)$$. Combining the theorem with other known methods gives the algorithm for a computation of the value of $$\rho(A)$$.

##### MSC:
 15A18 Eigenvalues, singular values, and eigenvectors 15B48 Positive matrices and their generalizations; cones of matrices 65F15 Numerical computation of eigenvalues and eigenvectors of matrices
##### Keywords:
Perron complement; spectral radius; Perron root; algorithm
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##### References:
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