Brief content of Numerical analysis post 10-linear.

10-Solved problems for 3×3 matrix by Crout’s LU-option 1.

Solved problems for 3×3 matrix by Crout’s LU-option -1.

We have two solved problems for the 3×3 matrix by Crout’s LU decomposition. for which we will apply the same technique used to derive the elements of the lower matrix and the upper matrix.

The video for first solved problem-Option-1.

The video used is for the illustration of the first solved problem of Crout’s LU decomposition for 3 x3 matrices. The solution is based on option-1, for which the matrix A is converted to a lower matrix to get the values for the different elements of the two matrices L and U.

You can click on any picture to enlarge then press the small arrow at the right to review all the other images as a slide show.

The first solved problem for 3×3 matrix by Crout’s LU-Option-1.

The first problem for the 3×3 matrix is by Crout’s LU decomposition. It is required to find both the Lower and upper matrix decomposition LU by using Crout’s method through a given matrix A. The given matrix is 3×3.

We will apply the previous post technique to get the L1 and U1 matrix components by using the following steps.

Step-1 Convert matrix A to a lower matrix.

First, we have matrix A as (1 1 1, 4 3 -1, 3 5 3 ). We will estimate the determinant value of this matrix A, it is estimated as equal to +10. Since the determinant value is not equal to zero, then the matrix is invertible and we can proceed to find L and U matrices based on Crout’s method.

 The three elements in the first column vector of the A matrix which are 1 4 3 will be set as L11 &L21 and L31. L11=1, L21=4,L31=3. While for U12, we will divide a21/a11=1/1=1. While U13=a13/a11=1.

The first solved problem of the two solved problems for 3x3 matrix by Crout's LU decomposition.

As for the upper matrix U1, the u11=1, U12=a21/a11=1/1=1 and u13=a13/a11=1/1=1. The detailed procedures are shown in the slide image.

Step-2-Derive the values for the L22, L23, and U23-option-1.

We will use a11 as a pivot, we multiply (-1)*(a12/a11)*column 1 and add the result to the 2nd column, similarly, we multiply (-1)*(a13/a11)*column 1 and add the result to the 3rd column.  If we call the new matrix L1, the elements of the matrix are as follows:

The first row is (1   0   0). The second row is (4  -1   -5). The third row is (3  2   0). If we compare the first row of L1 to the next matrix, we have (L11 0 0), then L11=1. If we compare the second row of L1 to the next matrix, we have (L21 L22  L22*U23), then L21=4.

If we compare the third row of L1 to the next matrix, we have (L31 L32  L32*U23+L33), then L31=3 and L32=2.

First solved problem -How to get the value of L22 for L1 matrix?

To get U23 divide (-5)/(-1)=5.

Step-3-Derive the value for the L33.

We want to proceed to get the final l matrix, we want to let  -5 to equal zero. We multiply (-1)*(-5/-1)*column 2 and add the result to the 3rd column. The final elements of the Lower matrix l will be as follows:

The first row is (1   0   0). The second row is (  4 -1  -5*-1+(-5)). The third row is (  3  2  -5*2+(0)).

1st solved problem-How to get the value of L33?

As we can see L33 will be equal to -10.

Step-4-Check the product of(L*U)=A-matrix for a 3×3 matrix.

In the final slide, we have all the elements in the first column of the matrix identical to the first column of the lower matrix. The diagonals of the U matrix U11=U22=U33=1. L22=-1. U23=+5 while L33=-10. U12=1 and u13=1.

We can write both the lower matrix and the upper matrix and check whether the multiplication of the lower matrix by the upper matrix will give us the final A matrix or not.

How to derive the expression of L and U matrices?

Using the row by column multiplication, we can see that L*U multiplication will give the matrix A.

The second solved problem-Crout’s LU-Option-1.

First, we have matrix A as ( 10 3 4, 2 – 10 3, 3 2 -10 ). We will estimate the determinant value of this matrix A, it is estimated as equal to +1163 Since the determinant value is not equal to zero, then the matrix is invertible and we can proceed to find L and U matrices based on Crout’s method.

2nd solved problem-check whether matrix is invertible.

The second solved problem, the detailed procedure to estimate the determinant is shown in the next slide image.

Step-1 Convert matrix A to a lower matrix.

The three elements in the first column vector of the A matrix which are 10 2 3 will be set as L11 &L21 and L31. L11=10, L21=2,L31=3. While for U12, we will divide a21/a11=3/10=3/10.

The second solved problem of the two solved problems for 3x3 matrix by Crout's LU decomposition.

While U13=a13/a11=4/10.

Step-2-Derive the values for the L22, L23, and U23.

We will use a11 as a pivot, we multiply (-1)*(a12/a11)*column 1 and add the result to the 2nd column, similarly, we multiply (-1)*(a13/a11)*column 1 and add the result to the 3rd column.  If we call the new matrix L1, the elements of the matrix are as follows:

The first row is (10 3 4). The second row is (2 -10 3). The third row is (3  2   -10). If we compare the first row of L1 to the next matrix, we have (L11 0 0), then L11=1. If we compare the second row of L1 to the next matrix, we have (L21 L22  L22*U23), then L21=2.

If we compare the third row of L1 to the next matrix, we have (L31 L32  L32*U23+L33), then L31=3 and L32=1.1.

Second solved problem.Derive the values for L22 and L32.

To get U23 divide ((2.2)/(-10.6)=-11/53.

Step-3-Derive the value for the L33-Crout’s LU decomposition.

We want to proceed to get the final l matrix, we want to let  2.2 to be equal zero. We multiply (-1)*(2.2/-10.60)*column 2 and add the result to the 3rd column. The final elements of the Lower matrix l will be as follows:

The first row is (10   0   0). The second row is (  2 -10.60 0). The third row is (  3  1.1 -1163/106).

The 2nd solved problem -Find the value of L33.

As we can see L33 will be equal to -1163/106

Step-4-Check the product of(L*U)=A-matrix for a 3×3 matrix.

2nd solved problem check L*u.

Using the row by column multiplication, we can see that L*U multiplication will give the matrix A.

The full detailed steps are shown in the previous slide images, at the end we will check the product of L*U against the Matrix A value.

This is a link to the next post is the introduction to the permutation matrix.

References

This is the Omni calculator for estimating various items of linear algebra -LU Decomposition Calculator.

Another calculator to use is Calculator for matrices.

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