Let /0 1 0 --6:0 A = 30 3 0 2 0 (a) Show that A is irreducible. (b) Find the Perron root, the right and the left Perron vectors of A. (c) What are the eigenvalues that are on the spectral circle of A.
Let /0 1 0 --6:0 A = 30 3 0 2 0 (a) Show that A is irreducible. (b) Find the Perron root, the right and the left Perron vectors of A. (c) What are the eigenvalues that are on the spectral circle of A.
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 question, thanks
![### Matrix Analysis Problem
1. **Given:**
\[
A = \begin{pmatrix}
0 & 1 & 0 \\
3 & 0 & 3 \\
0 & 2 & 0
\end{pmatrix}
\]
### Problems:
(a) **Show that \(A\) is irreducible.**
(b) **Find the Perron root, the right and the left Perron vectors of \(A\).**
(c) **What are the eigenvalues that are on the spectral circle of \(A\)?**
### Explanation:
(a) An irreducible matrix is one that cannot be transformed into a block upper triangular form by simultaneous row and column permutations. In other words, \(A\) is irreducible if for some reordering of \(A\), you cannot reach a form where there is a zero matrix in the bottom-left corner. You will need to show that for every pair of indices \( (i, j) \), there exists a positive integer \( k \) such that the \( i,j \)-entry of \( A^k \) is non-zero.
(b) The Perron-Frobenius theorem applies to irreducible matrices and asserts that there is a unique largest eigenvalue (Perron root) which is real and positive. You will need to compute the eigenvalues of \(A\) and identify this largest eigenvalue. The corresponding eigenvector is the right Perron vector, and you will also need to find the left eigenvector corresponding to this eigenvalue.
(c) The spectral circle of a matrix is the circle in the complex plane centered at the origin with radius equal to the Perron root. The eigenvalues lying on this circle are the roots of the characteristic polynomial of \(A\) that have a magnitude equal to the Perron root. Calculate the eigenvalues of \(A\) and determine which ones lie on this circle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F979d0aba-5428-414f-a3ba-5510f0301082%2F440bdbb0-c4af-4500-98cd-90fae007cca2%2Fm9vc1l_processed.png&w=3840&q=75)
Transcribed Image Text:### Matrix Analysis Problem
1. **Given:**
\[
A = \begin{pmatrix}
0 & 1 & 0 \\
3 & 0 & 3 \\
0 & 2 & 0
\end{pmatrix}
\]
### Problems:
(a) **Show that \(A\) is irreducible.**
(b) **Find the Perron root, the right and the left Perron vectors of \(A\).**
(c) **What are the eigenvalues that are on the spectral circle of \(A\)?**
### Explanation:
(a) An irreducible matrix is one that cannot be transformed into a block upper triangular form by simultaneous row and column permutations. In other words, \(A\) is irreducible if for some reordering of \(A\), you cannot reach a form where there is a zero matrix in the bottom-left corner. You will need to show that for every pair of indices \( (i, j) \), there exists a positive integer \( k \) such that the \( i,j \)-entry of \( A^k \) is non-zero.
(b) The Perron-Frobenius theorem applies to irreducible matrices and asserts that there is a unique largest eigenvalue (Perron root) which is real and positive. You will need to compute the eigenvalues of \(A\) and identify this largest eigenvalue. The corresponding eigenvector is the right Perron vector, and you will also need to find the left eigenvector corresponding to this eigenvalue.
(c) The spectral circle of a matrix is the circle in the complex plane centered at the origin with radius equal to the Perron root. The eigenvalues lying on this circle are the roots of the characteristic polynomial of \(A\) that have a magnitude equal to the Perron root. Calculate the eigenvalues of \(A\) and determine which ones lie on this circle.
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