ization of Apply modified Gram-Schmidt orthogonalization to find the full QR factor- [48 1 A 0 2 -2 36 7 H
ization of Apply modified Gram-Schmidt orthogonalization to find the full QR factor- [48 1 A 0 2 -2 36 7 H
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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![**Title: QR Factorization Using Modified Gram-Schmidt Orthogonalization**
**Objective:**
To find the full QR factorization of a given matrix using the modified Gram-Schmidt orthogonalization process.
**Problem Statement:**
Apply modified Gram-Schmidt orthogonalization to find the full QR factorization of the matrix:
\[
A = \begin{bmatrix}
4 & 8 & 1 \\
0 & 2 & -2 \\
3 & 6 & 7
\end{bmatrix}
\]
**Explanation:**
In the modified Gram-Schmidt process, the matrix \( A \) is decomposed into two matrices: \( Q \) and \( R \). Here, \( Q \) is an orthogonal matrix, and \( R \) is an upper triangular matrix. This method iteratively orthogonalizes the columns of \( A \) and ensures numerical stability compared to the classical Gram-Schmidt process.
**Procedure:**
1. **Initialization:** Start with the first column of \( A \).
2. **Orthogonalization:** Iteratively project each successive column against the previously computed orthogonal vectors.
3. **Normalization:** Divide each orthogonal vector by its norm to form the columns of \( Q \).
4. **Construction of \( R \):** The elements of \( R \) are determined during the orthogonalization process.
The process ensures that each column of \( Q \) maintains orthogonality with respect to the others, thus preserving the orthogonal invariants, while the matrix \( R \) captures the coefficients required to reconstruct \( A \) as \( A = QR \).
This approach is particularly useful in numerical linear algebra for solving linear systems, eigenvalue problems, and more, where maintaining numerical accuracy and stability is crucial.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8072c6db-e911-45d0-83c0-abe401e94f38%2Fed27add0-5894-46e4-9c05-60a6b14eca8a%2Fub9l5t_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: QR Factorization Using Modified Gram-Schmidt Orthogonalization**
**Objective:**
To find the full QR factorization of a given matrix using the modified Gram-Schmidt orthogonalization process.
**Problem Statement:**
Apply modified Gram-Schmidt orthogonalization to find the full QR factorization of the matrix:
\[
A = \begin{bmatrix}
4 & 8 & 1 \\
0 & 2 & -2 \\
3 & 6 & 7
\end{bmatrix}
\]
**Explanation:**
In the modified Gram-Schmidt process, the matrix \( A \) is decomposed into two matrices: \( Q \) and \( R \). Here, \( Q \) is an orthogonal matrix, and \( R \) is an upper triangular matrix. This method iteratively orthogonalizes the columns of \( A \) and ensures numerical stability compared to the classical Gram-Schmidt process.
**Procedure:**
1. **Initialization:** Start with the first column of \( A \).
2. **Orthogonalization:** Iteratively project each successive column against the previously computed orthogonal vectors.
3. **Normalization:** Divide each orthogonal vector by its norm to form the columns of \( Q \).
4. **Construction of \( R \):** The elements of \( R \) are determined during the orthogonalization process.
The process ensures that each column of \( Q \) maintains orthogonality with respect to the others, thus preserving the orthogonal invariants, while the matrix \( R \) captures the coefficients required to reconstruct \( A \) as \( A = QR \).
This approach is particularly useful in numerical linear algebra for solving linear systems, eigenvalue problems, and more, where maintaining numerical accuracy and stability is crucial.
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