2. The matrix A has eigenvalues -2,-2, and 1. Find 3 linearly independent eigenvec- tors for A. Show enough work to show you know how to do this problem by hand. 3 3
2. The matrix A has eigenvalues -2,-2, and 1. Find 3 linearly independent eigenvec- tors for A. Show enough work to show you know how to do this problem by hand. 3 3
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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Please help with questions 2 and 3.
![### Linear Algebra: Eigenvalues and Diagonalization
#### Problem 2
The matrix \( A \) has eigenvalues \(-2\), \(-2\), and \(1\). Find 3 linearly independent eigenvectors for \( A \). Show enough work to demonstrate your understanding of how to solve this problem by hand.
The matrix \( A \) is given by:
\[
A = \begin{bmatrix}
1 & 3 & 3 \\
-3 & -5 & -3 \\
3 & 3 & 1
\end{bmatrix}
\]
#### Solution Approach:
1. **Step 1: Eigenvalue Calculation:**
- The eigenvalues are provided: \(-2\), \(-2\), and \(1\).
2. **Step 2: Eigenvector Calculation for \( \lambda = -2 \):**
- Solve \((A - (-2)I) \mathbf{v} = 0\) to find the eigenvectors associated with \( \lambda = -2 \).
3. **Step 3: Eigenvector Calculation for \( \lambda = 1 \):**
- Solve \((A - I) \mathbf{v} = 0\) to find the eigenvectors associated with \( \lambda = 1 \).
#### Problem 3
Diagonalize the matrix in part 1a. Again, show enough work to demonstrate your understanding of how to solve this problem by hand.
#### Solution Approach:
1. **Step 1: Matrix of Eigenvectors \( P \):**
- Construct a matrix \( P \) where each column is an eigenvector of \( A \).
2. **Step 2: Diagonal Matrix \( D \):**
- Construct the diagonal matrix \( D \), where the diagonal elements are the eigenvalues of \( A \).
3. **Step 3: Verification:**
- Verify that \( A = PDP^{-1} \).
Ensure to demonstrate each step clearly and provide calculations to show your method and understanding.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F54d39984-15d1-49e5-8ad0-848d7106f454%2Ffb6f39a4-d4e7-43a1-b23d-483a7c9d1eac%2F7xtc8v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra: Eigenvalues and Diagonalization
#### Problem 2
The matrix \( A \) has eigenvalues \(-2\), \(-2\), and \(1\). Find 3 linearly independent eigenvectors for \( A \). Show enough work to demonstrate your understanding of how to solve this problem by hand.
The matrix \( A \) is given by:
\[
A = \begin{bmatrix}
1 & 3 & 3 \\
-3 & -5 & -3 \\
3 & 3 & 1
\end{bmatrix}
\]
#### Solution Approach:
1. **Step 1: Eigenvalue Calculation:**
- The eigenvalues are provided: \(-2\), \(-2\), and \(1\).
2. **Step 2: Eigenvector Calculation for \( \lambda = -2 \):**
- Solve \((A - (-2)I) \mathbf{v} = 0\) to find the eigenvectors associated with \( \lambda = -2 \).
3. **Step 3: Eigenvector Calculation for \( \lambda = 1 \):**
- Solve \((A - I) \mathbf{v} = 0\) to find the eigenvectors associated with \( \lambda = 1 \).
#### Problem 3
Diagonalize the matrix in part 1a. Again, show enough work to demonstrate your understanding of how to solve this problem by hand.
#### Solution Approach:
1. **Step 1: Matrix of Eigenvectors \( P \):**
- Construct a matrix \( P \) where each column is an eigenvector of \( A \).
2. **Step 2: Diagonal Matrix \( D \):**
- Construct the diagonal matrix \( D \), where the diagonal elements are the eigenvalues of \( A \).
3. **Step 3: Verification:**
- Verify that \( A = PDP^{-1} \).
Ensure to demonstrate each step clearly and provide calculations to show your method and understanding.
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