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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
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