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(a) Let A be a nonnegative irreducible matrix such that rz ≤ Az, for some 0 ‡ z ≥ 0, where r = p(A). Then show that rz = Az and z>0. (b) Let A be a nonnegative matrix with the property that each row sum is at most 1. Suppose further that at least one row sum is less than 1. Such a matrix is called a substochastic matrix. Using item (a), show that if A is an irreducible substochastic matrix, then p(A) < 1.

(a) Let A be a nonnegative irreducible matrix such that rz ≤ Az, for some 0 ‡ z ≥ 0, where r = p(A). Then show that rz = Az and z>0. (b) Let A be a nonnegative matrix with the property that each row sum is at most 1. Suppose further that at least one row sum is less than 1. Such a matrix is called a substochastic matrix. Using item (a), show that if A is an irreducible substochastic matrix, then p(A) < 1.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.1: Operations With Matrices
Problem 77E
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3. (a) Let A be a nonnegative irreducible matrix such that rz ≤ Az, for some 0 ‡z ≥ 0, where r = p(A). Then show that rz = Az and z> 0. (b) Let A be a nonnegative matrix with the property that each row sum is at most 1. Suppose further that at least one row sum is less than 1. Such a matrix is called a substochastic matrix. Using item (a), show that if A is an irreducible substochastic matrix, then p(A) < 1.
Transcribed Image Text:3. (a) Let A be a nonnegative irreducible matrix such that rz ≤ Az, for some 0 ‡z ≥ 0, where r = p(A). Then show that rz = Az and z> 0. (b) Let A be a nonnegative matrix with the property that each row sum is at most 1. Suppose further that at least one row sum is less than 1. Such a matrix is called a substochastic matrix. Using item (a), show that if A is an irreducible substochastic matrix, then p(A) < 1.
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