2. Find the LU factorization of the following matrix. 1 1 A = | 1 2 3 1 3 6 /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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### LU Factorization Problem
#### Problem Statement:
Find the LU factorization of the following matrix.
\[ A = \begin{pmatrix}
1 & 1 & 1 \\
1 & 2 & 3 \\
1 & 3 & 6
\end{pmatrix} \]
---
### Explanation:
LU factorization decomposes a matrix into the product of a lower triangular matrix \( L \) and an upper triangular matrix \( U \).
For the matrix \( A \) given above, we seek matrices \( L \) and \( U \) such that:
\[ A = LU \]
Where:
- \( L \) is a lower triangular matrix.
- \( U \) is an upper triangular matrix.
Begin with \( A \):
\[ A = \begin{pmatrix}
1 & 1 & 1 \\
1 & 2 & 3 \\
1 & 3 & 6
\end{pmatrix} \]
1. **Upper Triangular Matrix \( U \)**: It will have the form:
\[ U = \begin{pmatrix}
u_{11} & u_{12} & u_{13} \\
0 & u_{22} & u_{23} \\
0 & 0 & u_{33}
\end{pmatrix} \]
2. **Lower Triangular Matrix \( L \)**: It will have the form:
\[ L = \begin{pmatrix}
1 & 0 & 0 \\
l_{21} & 1 & 0 \\
l_{31} & l_{32} & 1
\end{pmatrix} \]
Next steps involve the elimination process to find the exact values of \( l_{ij} \) and \( u_{ij} \).
1. **Step 1**: Keep the first row of \( A \) as the first row of \( U \):
\[ U = \begin{pmatrix}
1 & 1 & 1 \\
0 & u_{22} & u_{23} \\
0 & 0 & u_{33}
\end{pmatrix} \]
2. **Step 2**: Eliminate the first column elements below \( u_{11} \):
\[ L = \begin{pmatrix}
1 & 0 & 0 \\
1 & 1 & 0 \\
1 & l_{32} &](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09b3997c-2096-4573-b74b-055c69181c0d%2Fb1a9c911-e106-4bae-a4a4-3a07427564c8%2F6ctpk9q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### LU Factorization Problem
#### Problem Statement:
Find the LU factorization of the following matrix.
\[ A = \begin{pmatrix}
1 & 1 & 1 \\
1 & 2 & 3 \\
1 & 3 & 6
\end{pmatrix} \]
---
### Explanation:
LU factorization decomposes a matrix into the product of a lower triangular matrix \( L \) and an upper triangular matrix \( U \).
For the matrix \( A \) given above, we seek matrices \( L \) and \( U \) such that:
\[ A = LU \]
Where:
- \( L \) is a lower triangular matrix.
- \( U \) is an upper triangular matrix.
Begin with \( A \):
\[ A = \begin{pmatrix}
1 & 1 & 1 \\
1 & 2 & 3 \\
1 & 3 & 6
\end{pmatrix} \]
1. **Upper Triangular Matrix \( U \)**: It will have the form:
\[ U = \begin{pmatrix}
u_{11} & u_{12} & u_{13} \\
0 & u_{22} & u_{23} \\
0 & 0 & u_{33}
\end{pmatrix} \]
2. **Lower Triangular Matrix \( L \)**: It will have the form:
\[ L = \begin{pmatrix}
1 & 0 & 0 \\
l_{21} & 1 & 0 \\
l_{31} & l_{32} & 1
\end{pmatrix} \]
Next steps involve the elimination process to find the exact values of \( l_{ij} \) and \( u_{ij} \).
1. **Step 1**: Keep the first row of \( A \) as the first row of \( U \):
\[ U = \begin{pmatrix}
1 & 1 & 1 \\
0 & u_{22} & u_{23} \\
0 & 0 & u_{33}
\end{pmatrix} \]
2. **Step 2**: Eliminate the first column elements below \( u_{11} \):
\[ L = \begin{pmatrix}
1 & 0 & 0 \\
1 & 1 & 0 \\
1 & l_{32} &
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