How to use the LU decomposition method for the 2×2 matrix?
LU decomposition method for the 2×2 matrix?
Now we will start The lower /upper decomposition of LU’s triangulation method.
The idea behind that is possible that any square matrix A can be expressed as a product of what is called lower triangular matrix L and Upper triangular Matrix U. For example if we have two sets of equations 2x + 3y = 13 and 3x + 4y = 18, which constitutes a matrix of 2×2, we are calling this matrix A.
This Matrix A We, Will, set it to be equal to two matrices multiplied by each other One of these is called the lower Matrix. The other one is called the upper matrix.
The lower matrix property is a matrix. That has a diagonal, which is not zero and the lower corner is having a value which is not zero while in the upper corner each element will be=0.
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On the contrary, the upper matrix is a matrix that has a diagonal of non zero and the upper corner is also non-zero, while the lower corner is =zero.
Just why this is called a lower matrix it is enclosed by a diagonal and lower bottom While the upper matrix is enclosed by the diagonal and the upper right-hand side.
For the symbols. thus the matrix can be decomposed by using the Lu decomposition method.
LU decomposition method by using an example of 2×2 matrix.
For the symbols, we will write A matrix (2 3 3 4, we are saying a11, a12, a21, and a22. We have six unknowns, which are L11,L21,l22,U11, U12 and U22.
Due the multiplication of the lower by the upper Matrix, we have only four equations.
According to the LU triangulation method, the two elements L11 and L22 will be=1. Hence we can solve the four equations to get the values of the remaining unknowns.
Since A = the lower multiplication by the upper matrices, we will perform the multiplication of lower and upper metrics and equate the product to the elements of matrix A. For example, if we multiply the first row by the First Column, we will get the value of 2, which we express as (1*U11+0)=2.
The multiplication of the First row by the second column will give us 3. While a22, which is equal to 4, will be equivalent to the multiplication of the second row by the second column,( L21 *U12)+(1*U22)=4. From equation I, we have U11=2.
From equation II, we have U12=3. Substitute the value of U11 in equation three and get the value of L21 is equal to 3/2.
Finally, we can get the value of u22 from equation four, after substituting the values of L21 and U12. U22 will be equal to -1/2.
Remaining for us the minus(-) (1/2) is denoted by U22, which is = 4 – 3, which is L21, 2 * 3 = -1/2. or U22=a22-L21*U12.
If we put the Matrix on the left-hand side and the corresponding values after solving the 4 equations, we will find an exciting remark that a11 will go to U11. And a12 will go to U12. As if we are taking these values. From here to there.
Consider a11 as a pivot divide a21/a11 which is equal to 3/2 will come to constitute a value of L21. This is from the equations which we have solved.
While multiply a21/a11 by minus 1 will give the value of U21.
Please refer to the detailed illustration in the following image.
Does all matrix 2×2 have LU decomposition?
Not all 2×2 matrix has Lu decomposition, for the case of determinant value=0, LU decomposition can not be considered because it will lead to a U matrix with zero value at the diagonal.
To estimate the values of elements of the upper Matrix, we have U11=a11=1, and U12 is equal to a22=5, by definition. U12=0. Finally, U22=a12-L21*u12=10*(2*5)=0. U22 cannot equal zero since U is an upper matrix with the diagonal elements being nonzeros.
This is the upper matrix for matrix B.
The expression of B=LU is not acceptable.
This is the pdf file used for the illustration of this post.
The next post, post 3, Best illustration-how to solve for x-y for two equations-L/U Decomposition?
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