Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D To save time, the eigenvalues are –4 and 0. - 4 - 2 - 2 A = - 4 - 2 0 - 2 Enter the matrices P and D below. (Use a comma to separate answers as needed. Type exact answers, using radicals as needed. Do not label the matrices.)
Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D To save time, the eigenvalues are –4 and 0. - 4 - 2 - 2 A = - 4 - 2 0 - 2 Enter the matrices P and D below. (Use a comma to separate answers as needed. Type exact answers, using radicals as needed. Do not label the matrices.)
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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![### Orthogonal Diagonalization of a Matrix
---
**Problem Statement:**
Orthogonally diagonalize the matrix, giving an orthogonal matrix \( P \) and a diagonal matrix \( D \). To save time, the eigenvalues are \( -4 \) and \( 0 \).
Given Matrix \( A \):
\[
A = \begin{pmatrix}
-4 & 0 & 0 & 0 \\
0 & -2 & 0 & -2 \\
0 & 0 & -4 & 0 \\
0 & -2 & 0 & -2
\end{pmatrix}
\]
---
**Task:**
Enter the matrices \( P \) and \( D \) below.
(Use a comma to separate answers as needed. Type exact answers, using radicals as needed. Do not label the matrices.)
---
- **Input Field:**
- This is where you will type the orthogonal matrix \( P \) and the diagonal matrix \( D \).
---
In orthogonal diagonalization, the goal is to find a matrix \( P \) that satisfies \( P^TAP = D \), where \( D \) is a diagonal matrix consisting of the eigenvalues of \( A \), and \( P \) is an orthogonal matrix whose columns are the normalized eigenvectors of \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9225fa68-a539-4743-ab4b-10a3968e5a82%2Fe3771b16-7aba-489f-845b-3a6d35c919b2%2Fiiwpc8d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Orthogonal Diagonalization of a Matrix
---
**Problem Statement:**
Orthogonally diagonalize the matrix, giving an orthogonal matrix \( P \) and a diagonal matrix \( D \). To save time, the eigenvalues are \( -4 \) and \( 0 \).
Given Matrix \( A \):
\[
A = \begin{pmatrix}
-4 & 0 & 0 & 0 \\
0 & -2 & 0 & -2 \\
0 & 0 & -4 & 0 \\
0 & -2 & 0 & -2
\end{pmatrix}
\]
---
**Task:**
Enter the matrices \( P \) and \( D \) below.
(Use a comma to separate answers as needed. Type exact answers, using radicals as needed. Do not label the matrices.)
---
- **Input Field:**
- This is where you will type the orthogonal matrix \( P \) and the diagonal matrix \( D \).
---
In orthogonal diagonalization, the goal is to find a matrix \( P \) that satisfies \( P^TAP = D \), where \( D \) is a diagonal matrix consisting of the eigenvalues of \( A \), and \( P \) is an orthogonal matrix whose columns are the normalized eigenvectors of \( A \).
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