Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### LU Decomposition (Factorization) - Educational Content
#### LU Decomposition
LU decomposition involves factoring a matrix \( A \) into the product of a lower triangular matrix \( L \) and an upper triangular matrix \( U \).
#### Problem Statement
Find an LU decomposition (factorization) of:
\[
A = \begin{bmatrix}
8 & 9 \\
-5 & 3
\end{bmatrix}
\]
That is, write \( A = LU \) where \( L \) is a lower triangular matrix with ones in the diagonal, and \( U \) is an upper triangular matrix.
#### Application of LU Decomposition
Use the LU factorization to solve the system:
\[
\begin{bmatrix}
8 & 9 \\
-5 & 3
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
10 \\
11
\end{bmatrix}
\]
#### Step-by-Step Solution Process
1. **LU Factorization:**
\[
A = LU = \begin{bmatrix}
1 & 0 \\
l_{21} & 1
\end{bmatrix}
\begin{bmatrix}
u_{11} & u_{12} \\
0 & u_{22}
\end{bmatrix}
\]
2. **Construct matrices \( L \) and \( U \) such that:**
\[
\begin{bmatrix}
8 & 9 \\
-5 & 3
\end{bmatrix} =
\begin{bmatrix}
1 & 0 \\
l_{21} & 1
\end{bmatrix}
\begin{bmatrix}
u_{11} & u_{12} \\
0 & u_{22}
\end{bmatrix}
\]
- From the first row: \(u_{11} = 8\) and \( u_{12} = 9 \).
- From the element \( -5 = l_{21} \times 8 \) we get \( l_{21} = -\frac{5}{8}](https://content.bartleby.com/qna-images/question/088399be-9a43-44b3-a949-bec438b8f98b/015674c8-0a31-4bdc-851f-7f489ae8f156/0s78v9i_processed.jpeg)
Transcribed Image Text:### LU Decomposition (Factorization) - Educational Content
#### LU Decomposition
LU decomposition involves factoring a matrix \( A \) into the product of a lower triangular matrix \( L \) and an upper triangular matrix \( U \).
#### Problem Statement
Find an LU decomposition (factorization) of:
\[
A = \begin{bmatrix}
8 & 9 \\
-5 & 3
\end{bmatrix}
\]
That is, write \( A = LU \) where \( L \) is a lower triangular matrix with ones in the diagonal, and \( U \) is an upper triangular matrix.
#### Application of LU Decomposition
Use the LU factorization to solve the system:
\[
\begin{bmatrix}
8 & 9 \\
-5 & 3
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
10 \\
11
\end{bmatrix}
\]
#### Step-by-Step Solution Process
1. **LU Factorization:**
\[
A = LU = \begin{bmatrix}
1 & 0 \\
l_{21} & 1
\end{bmatrix}
\begin{bmatrix}
u_{11} & u_{12} \\
0 & u_{22}
\end{bmatrix}
\]
2. **Construct matrices \( L \) and \( U \) such that:**
\[
\begin{bmatrix}
8 & 9 \\
-5 & 3
\end{bmatrix} =
\begin{bmatrix}
1 & 0 \\
l_{21} & 1
\end{bmatrix}
\begin{bmatrix}
u_{11} & u_{12} \\
0 & u_{22}
\end{bmatrix}
\]
- From the first row: \(u_{11} = 8\) and \( u_{12} = 9 \).
- From the element \( -5 = l_{21} \times 8 \) we get \( l_{21} = -\frac{5}{8}
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- What is the Matrix multiplication?arrow_forwardTransform the below system into the simpler upper triangular form & describ in terms of elementary row operations on the augmented coefficient matrix a11x1 + a12x2 +. . . + a1nxn = b1,a21x1 + a22x2 +" " " + a2nxn = b2,an1x1 + an2x2 +" " " + annxn =bnarrow_forwardb11 b21 Multiply the following two matrices to obtain the product matrix B. = = b12 b22 = = 6483-6] }] = [b21 b22 b₁1 b12arrow_forward
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