Find a QR factorization of matrix A = -1 0 20 2 4 -4 3 6 -6 -4 0 0 6 -8 0 15
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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![## QR Factorization of a Matrix
### Problem Statement
Find a QR factorization of matrix \( A \):
\[ A = \begin{bmatrix} -1 & 2 & 3 & 4 \\ 0 & 4 & 6 & 6 \\ 2 & -4 & -6 & -8 \\ 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 15 \end{bmatrix} \]
### Steps to Solve
1. **Use the Gram-Schmidt Orthogonalization Algorithm**:
- Find an orthogonal basis for the column space of matrix \( A \).
2. **Normalize Vectors**:
- Normalize the vectors obtained in the previous step.
3. **Form Matrix \( Q \)**:
- Use the normalized vectors to form matrix \( Q \).
4. **Express Matrix \( R \)**:
- Derive the upper triangular matrix \( R \) using matrices \( A \) and \( Q \).
5. **Find Entries of \( R \)**:
- Calculate the specific values for the entries in matrix \( R \).
This process will result in matrices \( Q \) and \( R \) such that \( A = QR \), where \( Q \) is an orthogonal matrix and \( R \) is an upper triangular matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb73e221-fc24-4f22-8ff5-c1a81fef495f%2F574026ad-9b52-4cea-b1d8-a31de036da47%2Fvlt56w8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## QR Factorization of a Matrix
### Problem Statement
Find a QR factorization of matrix \( A \):
\[ A = \begin{bmatrix} -1 & 2 & 3 & 4 \\ 0 & 4 & 6 & 6 \\ 2 & -4 & -6 & -8 \\ 0 & -4 & 0 & 0 \\ 0 & 0 & 0 & 15 \end{bmatrix} \]
### Steps to Solve
1. **Use the Gram-Schmidt Orthogonalization Algorithm**:
- Find an orthogonal basis for the column space of matrix \( A \).
2. **Normalize Vectors**:
- Normalize the vectors obtained in the previous step.
3. **Form Matrix \( Q \)**:
- Use the normalized vectors to form matrix \( Q \).
4. **Express Matrix \( R \)**:
- Derive the upper triangular matrix \( R \) using matrices \( A \) and \( Q \).
5. **Find Entries of \( R \)**:
- Calculate the specific values for the entries in matrix \( R \).
This process will result in matrices \( Q \) and \( R \) such that \( A = QR \), where \( Q \) is an orthogonal matrix and \( R \) is an upper triangular matrix.
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