Find the matrix A that has the given eigenvalues and corresponding eigenvectors. 11 = 3> = 1 = A =
Find the matrix A that has the given eigenvalues and corresponding eigenvectors. 11 = 3> = 1 = 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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Question
![**Problem Statement:**
Find the matrix \( A \) that has the given eigenvalues and corresponding eigenvectors.
**Given:**
Eigenvalue \( \lambda_1 = 3 \) with the eigenvector \(\begin{bmatrix} 4 \\ 11 \end{bmatrix}\),
Eigenvalue \( \lambda_2 = 1 \) with the eigenvector \(\begin{bmatrix} 1 \\ 3 \end{bmatrix}\).
**Matrix Form:**
\[ A = \begin{bmatrix} \quad & \quad \\ \quad & \quad \end{bmatrix} \]
**Explanation:**
To find matrix \( A \), you will use the given eigenvalues and eigenvectors. A matrix can be constructed by using the formula:
\[ A = PDP^{-1} \]
where \( P \) is a matrix formed by placing the eigenvectors as columns, and \( D \) is a diagonal matrix formed by placing the eigenvalues along the diagonal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fef36e706-47e8-4994-a8d4-317113440949%2Faaad45f6-bee2-48bc-9c4d-89fd8cef101e%2F4g3gof_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the matrix \( A \) that has the given eigenvalues and corresponding eigenvectors.
**Given:**
Eigenvalue \( \lambda_1 = 3 \) with the eigenvector \(\begin{bmatrix} 4 \\ 11 \end{bmatrix}\),
Eigenvalue \( \lambda_2 = 1 \) with the eigenvector \(\begin{bmatrix} 1 \\ 3 \end{bmatrix}\).
**Matrix Form:**
\[ A = \begin{bmatrix} \quad & \quad \\ \quad & \quad \end{bmatrix} \]
**Explanation:**
To find matrix \( A \), you will use the given eigenvalues and eigenvectors. A matrix can be constructed by using the formula:
\[ A = PDP^{-1} \]
where \( P \) is a matrix formed by placing the eigenvectors as columns, and \( D \) is a diagonal matrix formed by placing the eigenvalues along the diagonal.
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