Find a Matrix P that Diagonalizes A, and check your answer by computing PDP-1 Be sure to give the Characteristic equation for the matrix and show the reduced matrix for the computation of each eigenvector. You may relegate all computations, such as reduction of the matrices and roots of the characteristic equation to the calculator 0 1
Find a Matrix P that Diagonalizes A, and check your answer by computing PDP-1 Be sure to give the Characteristic equation for the matrix and show the reduced matrix for the computation of each eigenvector. You may relegate all computations, such as reduction of the matrices and roots of the characteristic equation to the calculator 0 1
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
![Educational Task: Diagonalizing a Matrix via Eigenvectors and Eigenvalues
Objective: Determine the matrix \(P\) that diagonalizes \(A\) and verify your solution by computing \(PDP^{-1}\).
Instruction:
1. **Find the Characteristic Equation:**
- The characteristic equation of a matrix \(A\) is found by solving the determinant \( \text{det}(A - \lambda I) = 0 \), where \(\lambda\) represents the eigenvalues.
2. **Compute the Eigenvalues:**
- Solve the characteristic equation to find the eigenvalues of the matrix \(A\).
3. **Find the Eigenvectors:**
- For each eigenvalue \(\lambda\), solve the system \( (A - \lambda I) \mathbf{v} = 0 \) to find the corresponding eigenvectors.
4. **Form the Matrix \(P\) and \(D\):**
- Construct \(P\) using the eigenvectors as columns.
- Construct the diagonal matrix \(D\) using the eigenvalues.
5. **Verify by Multiplication:**
- Multiply \( PDP^{-1} \) to ensure that the original matrix \(A\) is obtained.
Given Matrix \(A\):
\[
A = \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
3 & 0 & 1
\end{pmatrix}
\]
Detailed Steps:
1. **Compute the Characteristic Polynomial:**
To find the characteristic polynomial, determine the determinant of \(A - \lambda I\):
\[
\text{det}(A - \lambda I) = \begin{vmatrix}
-\lambda & 0 & 0 \\
0 & -\lambda & 0 \\
3 & 0 & 1-\lambda
\end{vmatrix}
\]
2. **Reduction and Computation:**
Perform row and column operations and calculate the roots of the characteristic polynomial.
3. **Set Up the Eigenvalue Equation:**
Solve for eigenvectors based on each root (eigenvalue) by setting up the system of linear equations \( (A - \lambda I) \mathbf{v} = 0 \).
4. **Construct Matrices \(P\) and \(D\):**
Matrix \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5483afab-7850-40f6-959f-e698dea1419a%2Fbef25b55-fd91-46a6-b05f-30a6bd3b45be%2Frauo6wg_processed.png&w=3840&q=75)
Transcribed Image Text:Educational Task: Diagonalizing a Matrix via Eigenvectors and Eigenvalues
Objective: Determine the matrix \(P\) that diagonalizes \(A\) and verify your solution by computing \(PDP^{-1}\).
Instruction:
1. **Find the Characteristic Equation:**
- The characteristic equation of a matrix \(A\) is found by solving the determinant \( \text{det}(A - \lambda I) = 0 \), where \(\lambda\) represents the eigenvalues.
2. **Compute the Eigenvalues:**
- Solve the characteristic equation to find the eigenvalues of the matrix \(A\).
3. **Find the Eigenvectors:**
- For each eigenvalue \(\lambda\), solve the system \( (A - \lambda I) \mathbf{v} = 0 \) to find the corresponding eigenvectors.
4. **Form the Matrix \(P\) and \(D\):**
- Construct \(P\) using the eigenvectors as columns.
- Construct the diagonal matrix \(D\) using the eigenvalues.
5. **Verify by Multiplication:**
- Multiply \( PDP^{-1} \) to ensure that the original matrix \(A\) is obtained.
Given Matrix \(A\):
\[
A = \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
3 & 0 & 1
\end{pmatrix}
\]
Detailed Steps:
1. **Compute the Characteristic Polynomial:**
To find the characteristic polynomial, determine the determinant of \(A - \lambda I\):
\[
\text{det}(A - \lambda I) = \begin{vmatrix}
-\lambda & 0 & 0 \\
0 & -\lambda & 0 \\
3 & 0 & 1-\lambda
\end{vmatrix}
\]
2. **Reduction and Computation:**
Perform row and column operations and calculate the roots of the characteristic polynomial.
3. **Set Up the Eigenvalue Equation:**
Solve for eigenvectors based on each root (eigenvalue) by setting up the system of linear equations \( (A - \lambda I) \mathbf{v} = 0 \).
4. **Construct Matrices \(P\) and \(D\):**
Matrix \(
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