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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Matrix Analysis Exercise** **Problem 3:** **(a)** Let \( A \) be a nonnegative irreducible matrix such that \( rz \leq Az \), for some \( 0 \neq z \geq 0 \), where \( r = \rho(A) \). 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 \( \rho(A) < 1 \). --- **Explanation:** - **Irreducible Matrix**: An irreducible matrix is one that cannot be transformed into a block upper triangular matrix by simultaneous row and column permutations. This implies a certain connectivity in the graph-theoretic interpretation of the matrix. - **Nonnegative Matrix**: A matrix is nonnegative if all its entries are nonnegative (i.e., \( A_{ij} \geq 0 \) for all \( i, j \)). - **Spectral Radius** \( (\rho(A)) \): The spectral radius of a matrix \( A \) is the largest absolute value of its eigenvalues. - **Substochastic Matrix**: A substochastic matrix is a square matrix in which each row sums to at most 1, and at least one row sums to less than 1. --- **Solution Steps:** _For Part (a):_ 1. Start with the assumption that \( A \) is a nonnegative irreducible matrix, \( rz \leq Az \) and \( r = \rho(A) \). 2. Show that for the equality \( rz = Az \) to hold, \( z \) needs to be a positive vector. _For Part (b):_ 1. Given that \( A \) is a substochastic matrix, prove that under the conditions described, if \( A \) is also irreducible, its spectral radius \( \rho(A) \) must be less than 1. 2. Utilize the results and properties derived from (a) to establish the required proof for the spectral radius. This problem is a fundamental exercise in linear algebra and matrix theory,
Transcribed Image Text:**Matrix Analysis Exercise** **Problem 3:** **(a)** Let \( A \) be a nonnegative irreducible matrix such that \( rz \leq Az \), for some \( 0 \neq z \geq 0 \), where \( r = \rho(A) \). 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 \( \rho(A) < 1 \). --- **Explanation:** - **Irreducible Matrix**: An irreducible matrix is one that cannot be transformed into a block upper triangular matrix by simultaneous row and column permutations. This implies a certain connectivity in the graph-theoretic interpretation of the matrix. - **Nonnegative Matrix**: A matrix is nonnegative if all its entries are nonnegative (i.e., \( A_{ij} \geq 0 \) for all \( i, j \)). - **Spectral Radius** \( (\rho(A)) \): The spectral radius of a matrix \( A \) is the largest absolute value of its eigenvalues. - **Substochastic Matrix**: A substochastic matrix is a square matrix in which each row sums to at most 1, and at least one row sums to less than 1. --- **Solution Steps:** _For Part (a):_ 1. Start with the assumption that \( A \) is a nonnegative irreducible matrix, \( rz \leq Az \) and \( r = \rho(A) \). 2. Show that for the equality \( rz = Az \) to hold, \( z \) needs to be a positive vector. _For Part (b):_ 1. Given that \( A \) is a substochastic matrix, prove that under the conditions described, if \( A \) is also irreducible, its spectral radius \( \rho(A) \) must be less than 1. 2. Utilize the results and properties derived from (a) to establish the required proof for the spectral radius. This problem is a fundamental exercise in linear algebra and matrix theory,
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