LU Decomposition of A: Find Solution

In summary: Therefore, it is important to follow the necessary row operations, including interchanges, when finding the LU decomposition. In summary, the LU decomposition is a method for factoring a matrix into two separate matrices, L and U, in order to solve systems of linear equations. Row interchanges are necessary in order to get the matrix into the proper form for the decomposition, and the resulting U matrix may not be in the proper form without these interchanges. It is important to follow the necessary row operations, including interchanges, when finding the LU decomposition.
  • #1
jesuslovesu
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Homework Statement


A = [tex]
\left| \begin{array}{ccc}
2 & 0 & 4 \\
0 & 4 & -8 \\
3 & -1 & 2 \end{array} \right|.\]
[/tex]

Find the LU decomposition.

The Attempt at a Solution


My question is why do you need to interchange rows 3 & 1 at the start of the problem? My initial attempt was to find the upper matrix using the format as given. I got
U = [tex]
\left| \begin{array}{ccc}
2 & 0 & 4 \\
0 & 4 & -8 \\
0 & 0 & -6 \end{array} \right|.\]
[/tex]

doesn't that satisfy the requirements for an U matrix?

According to the calculator I'm using the U matrix should be
[tex]
\left| \begin{array}{ccc}
3 & -1 & 2 \\
0 & 4 & -8 \\
0 & 0 & 4 \end{array} \right|.\]
[/tex]
 
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  • #2


The LU decomposition is a way to factor a matrix into two separate matrices, L and U, where L is a lower triangular matrix and U is an upper triangular matrix. The purpose of this decomposition is to simplify the process of solving systems of linear equations. In order to find the LU decomposition of matrix A, it is necessary to perform row operations, including row interchanges, in order to get the matrix into the proper form.

In this case, the row interchange was necessary in order to get the matrix into the form of an upper triangular matrix. Without this interchange, the matrix would not have been in the proper form for the LU decomposition. The calculator is correct in showing the U matrix as \left| \begin{array}{ccc}
3 & -1 & 2 \\
0 & 4 & -8 \\
0 & 0 & 4 \end{array} \right|.\] because it takes into account the row interchange that was necessary to get the matrix into the proper form.

It is important to note that the LU decomposition is not unique, meaning that there can be multiple ways to factor a matrix into L and U matrices. However, the row interchanges are necessary in order to get the matrix into the standard form for the LU decomposition. Without these interchanges, the resulting U matrix may not be in the proper form.
 

1. What is LU Decomposition and why is it useful?

LU Decomposition is a method used to factorize a square matrix into a lower triangular matrix (L) and an upper triangular matrix (U). It is useful because it simplifies the process of solving systems of linear equations and can also be used for matrix inversion, which is important in many scientific and engineering applications.

2. How do you find the LU Decomposition of a matrix?

The LU Decomposition of a matrix A can be found by using an algorithm such as Gaussian elimination or Crout's method. These methods involve performing row operations on the matrix to reduce it to its upper triangular form. The resulting upper triangular matrix is U, and the lower triangular matrix L is constructed from the row operations used.

3. Can LU Decomposition be applied to any matrix?

No, LU Decomposition can only be applied to square matrices that are non-singular (have a non-zero determinant). This means that the matrix must have an inverse, which is necessary for solving systems of linear equations. If the matrix is singular, then LU Decomposition cannot be performed.

4. What is the difference between LU Decomposition and Gaussian elimination?

Gaussian elimination is a specific algorithm used to perform LU Decomposition, while LU Decomposition is a more general method for factorizing matrices. Gaussian elimination involves performing row operations on the matrix, while LU Decomposition involves finding two matrices, L and U, that when multiplied together equal the original matrix A.

5. How can LU Decomposition be used to solve systems of linear equations?

Once the LU Decomposition of a matrix A has been found, it can be used to solve systems of linear equations by first solving Ly = b for y, where L is the lower triangular matrix and b is the vector of constants. Then, Ux = y can be solved for x, where U is the upper triangular matrix and x is the vector of unknowns. This process is known as forward and backward substitution and is much simpler than solving the original system using Gaussian elimination.

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