Find an LU factorization of the following matrix (with L unit lower triangular). 1 3 -5 -31 8 4 -1 -5 A 4 -5 -7 -2 -4 7 5
Find an LU factorization of the following matrix (with L unit lower triangular). 1 3 -5 -31 8 4 -1 -5 A 4 -5 -7 -2 -4 7 5
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
![**LU Factorization of a Matrix**
**Problem Statement:**
Find an LU factorization of the following matrix \( A \) (with \( L \) unit lower triangular).
\[
A = \begin{pmatrix}
1 & 3 & -5 & -3 \\
-1 & -5 & 8 & 4 \\
4 & 2 & -5 & -7 \\
-2 & -4 & 7 & 5
\end{pmatrix}
\]
**Explanation:**
Given the matrix \( A \), the task is to decompose it into a product of a lower triangular matrix \( L \) (with unit diagonal) and an upper triangular matrix \( U \). This type of factorization is known as LU decomposition. In mathematical terms, we need to find matrices \( L \) and \( U \) such that:
\[
A = LU
\]
where:
- \( L \) is a lower triangular matrix with diagonal elements all equal to 1 (unit lower triangular),
- \( U \) is an upper triangular matrix.
To solve this, we follow the steps for LU decomposition which involve Gaussian elimination. The intermediate matrices constructed during elimination can be used to form the \( L \) and \( U \) matrices.
**Note:** Computation steps are omitted for brevity and focus on explanation. Please refer to a detailed mathematical resource for the step-by-step procedure of Gaussian elimination and LU factorization.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff869c4ac-8e58-4431-846d-0202389eaebe%2Fb92a9d74-e588-4ba8-9ca7-1504815571b3%2F8yn1g2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**LU Factorization of a Matrix**
**Problem Statement:**
Find an LU factorization of the following matrix \( A \) (with \( L \) unit lower triangular).
\[
A = \begin{pmatrix}
1 & 3 & -5 & -3 \\
-1 & -5 & 8 & 4 \\
4 & 2 & -5 & -7 \\
-2 & -4 & 7 & 5
\end{pmatrix}
\]
**Explanation:**
Given the matrix \( A \), the task is to decompose it into a product of a lower triangular matrix \( L \) (with unit diagonal) and an upper triangular matrix \( U \). This type of factorization is known as LU decomposition. In mathematical terms, we need to find matrices \( L \) and \( U \) such that:
\[
A = LU
\]
where:
- \( L \) is a lower triangular matrix with diagonal elements all equal to 1 (unit lower triangular),
- \( U \) is an upper triangular matrix.
To solve this, we follow the steps for LU decomposition which involve Gaussian elimination. The intermediate matrices constructed during elimination can be used to form the \( L \) and \( U \) matrices.
**Note:** Computation steps are omitted for brevity and focus on explanation. Please refer to a detailed mathematical resource for the step-by-step procedure of Gaussian elimination and LU factorization.
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