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Let A > 0 be such that each row (or column) sum of A equals r. Show that p(A) = r

Let A > 0 be such that each row (or column) sum of A equals r. Show that p(A) = r

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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### Problem Statement 4. Let \(A > 0\) be such that each row (or column) sum of \(A\) equals \(r\). Show that \(\rho(A) = r\). --- ### Detailed Explanation This problem deals with linear algebra and matrix properties. The task is to demonstrate that for a matrix \(A\) wherein each row or column sums to \(r\) (assuming all elements in \(A\) are greater than zero), the spectral radius \(\rho(A)\) of the matrix is equal to \(r\). Here are some key points related to the problem that might be discussed in an educational context: - **Spectral Radius**: The spectral radius of a matrix \(A\) is the largest absolute value of its eigenvalues. - **Matrix Row and Column Sum**: Each row sum (or column sum) being a constant means that the sum of the elements in each row (or each column) of the matrix \(A\) will be equal to the constant \(r\). #### Hints and Steps for Solution: 1. **Row Sum Equality**: Demonstrate that the sum of the entries in each row results in the same value \(r\). 2. **Using Perron-Frobenius Theorem**: Since \(A\) is a positive matrix (with all elements greater than zero), the Perron-Frobenius theorem can be applied, implying that \(A\) has a positive real eigenvalue that is larger or equal in absolute value to all other eigenvalues. 3. **Eigenvector Analysis**: Show that \(A\) has a positive eigenvector corresponding to the eigenvalue \(r\). 4. **Conclusion**: Conclude that the spectral radius, which is the largest eigenvalue, equals \(r\). Understanding these concepts is crucial for proving the problem statement. --- This explanation would be suitable for inclusion on an educational website that focuses on advanced mathematics, particularly linear algebra.
Transcribed Image Text:### Problem Statement 4. Let \(A > 0\) be such that each row (or column) sum of \(A\) equals \(r\). Show that \(\rho(A) = r\). --- ### Detailed Explanation This problem deals with linear algebra and matrix properties. The task is to demonstrate that for a matrix \(A\) wherein each row or column sums to \(r\) (assuming all elements in \(A\) are greater than zero), the spectral radius \(\rho(A)\) of the matrix is equal to \(r\). Here are some key points related to the problem that might be discussed in an educational context: - **Spectral Radius**: The spectral radius of a matrix \(A\) is the largest absolute value of its eigenvalues. - **Matrix Row and Column Sum**: Each row sum (or column sum) being a constant means that the sum of the elements in each row (or each column) of the matrix \(A\) will be equal to the constant \(r\). #### Hints and Steps for Solution: 1. **Row Sum Equality**: Demonstrate that the sum of the entries in each row results in the same value \(r\). 2. **Using Perron-Frobenius Theorem**: Since \(A\) is a positive matrix (with all elements greater than zero), the Perron-Frobenius theorem can be applied, implying that \(A\) has a positive real eigenvalue that is larger or equal in absolute value to all other eigenvalues. 3. **Eigenvector Analysis**: Show that \(A\) has a positive eigenvector corresponding to the eigenvalue \(r\). 4. **Conclusion**: Conclude that the spectral radius, which is the largest eigenvalue, equals \(r\). Understanding these concepts is crucial for proving the problem statement. --- This explanation would be suitable for inclusion on an educational website that focuses on advanced mathematics, particularly linear algebra.
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