Determine whether or not the following matrices have a Cholesky factorization; if they Ho, compute (by hand) the Cholesky factor R: A = [ 1 -2 0 -20 13 6 65 " B = 4 -4 0 -4 0 40 05
Determine whether or not the following matrices have a Cholesky factorization; if they Ho, compute (by hand) the Cholesky factor R: A = [ 1 -2 0 -20 13 6 65 " B = 4 -4 0 -4 0 40 05
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
![**Title: Cholesky Factorization of Matrices**
**Objective:**
Determine whether the following matrices have a Cholesky factorization; if they do, compute (by hand) the Cholesky factor \( R \).
**Matrix A:**
\[
A = \begin{bmatrix}
1 & -2 & 0 \\
-2 & 13 & 6 \\
0 & 6 & 5
\end{bmatrix}
\]
**Matrix B:**
\[
B = \begin{bmatrix}
4 & -4 & 0 \\
-4 & 4 & 0 \\
0 & 0 & 5
\end{bmatrix}
\]
**Explanation:**
To find if a matrix has a Cholesky factorization, it needs to be positive definite. If it is, the Cholesky factor \( R \) is an upper triangular matrix such that \( A = R^T R \).
1. **Positive Definite Check:**
- Ensure all leading principal minors of the matrix are positive.
2. **Cholesky Factorization:**
- If positive definite, proceed to compute the elements of \( R \) using the relations:
- \( R_{11} = \sqrt{A_{11}} \)
- \( R_{1j} = \frac{A_{1j}}{R_{11}} \) (for \( j > 1 \))
- \( R_{ii} = \sqrt{A_{ii} - \sum_{k=1}^{i-1} R_{ki}^2} \) (for \( i > 1 \))
- \( R_{ij} = \frac{A_{ij} - \sum_{k=1}^{i-1} R_{ki} R_{kj}}{R_{ii}} \) (for \( i < j \))](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed3eef2a-70e8-485e-b91d-563de65921b3%2F85c20e06-afc9-4f50-9460-96391bc89261%2Figv9icm_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Cholesky Factorization of Matrices**
**Objective:**
Determine whether the following matrices have a Cholesky factorization; if they do, compute (by hand) the Cholesky factor \( R \).
**Matrix A:**
\[
A = \begin{bmatrix}
1 & -2 & 0 \\
-2 & 13 & 6 \\
0 & 6 & 5
\end{bmatrix}
\]
**Matrix B:**
\[
B = \begin{bmatrix}
4 & -4 & 0 \\
-4 & 4 & 0 \\
0 & 0 & 5
\end{bmatrix}
\]
**Explanation:**
To find if a matrix has a Cholesky factorization, it needs to be positive definite. If it is, the Cholesky factor \( R \) is an upper triangular matrix such that \( A = R^T R \).
1. **Positive Definite Check:**
- Ensure all leading principal minors of the matrix are positive.
2. **Cholesky Factorization:**
- If positive definite, proceed to compute the elements of \( R \) using the relations:
- \( R_{11} = \sqrt{A_{11}} \)
- \( R_{1j} = \frac{A_{1j}}{R_{11}} \) (for \( j > 1 \))
- \( R_{ii} = \sqrt{A_{ii} - \sum_{k=1}^{i-1} R_{ki}^2} \) (for \( i > 1 \))
- \( R_{ij} = \frac{A_{ij} - \sum_{k=1}^{i-1} R_{ki} R_{kj}}{R_{ii}} \) (for \( i < j \))
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