Let A is nonnegative irreducible matrix such that I -A is invertible and (I -A)-1 >= 0. Let λ1 and v1 be the Perron root and Perron eigenvector of A. (a) Prove that 0 < λ1 < 1. (b) Deduce from (a) that v1 > Av1.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
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Let A is nonnegative irreducible matrix such that I -A is invertible and (I -A)-1 >= 0. Let λ1 and v1 be the Perron root and Perron eigenvector of A. (a) Prove that 0 < λ1 < 1. (b) Deduce from (a) that v1 > Av1.

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