The Lu Decomposition Calculator is used to factorize a square matrix with three rows and three columns into two matrices.
It decomposes a square matrix A into a lower triangular matrix L and an upper triangular matrix U.
The calculator takes a square matrix A with the order 3 x 3 as input and outputs the LU decomposition of the matrix that is the product of the matrices L and U. So, the matrix A can be written as:
A = LU
Where L and U are the lower triangular form and the upper triangular form of the square matrix A respectively. They both are special types of square matrices.
The lower triangular matrix is specified by having all the entries equal to zero which are above the main diagonal. Similarly, the upper triangular matrix has all the elements below its main diagonal equal to zero.
In LU decomposition, the entries above the main diagonal in the lower triangular matrix and the entries below the main diagonal in the upper triangular matrix are not altered.
The calculator only changes the remaining entries according to the matrix A.
The user can use this calculator to solve a system of three linear equations using LU decomposition. The coefficients in the system of three linear equations can be written in matrix form as:
AX = B
Where X is the unknown matrix. In LU decomposition, the matrix A is replaced with the product of matrices LU as follows:
LUX = B
The matrices L and U will be obtained by using this calculator. If we suppose UX=Y and substitute in the above equation, it gives:
LY = B
First solving for Y in the above equation and then putting the values of Y in UX = Y and then solving for X gives the solution of the system of three linear equations using LU decomposition.
What Is a LU Decomposition Calculator?
The Lu Decomposition Calculator is an online tool that is used to decompose a 3 x 3 square matrix A into the product of an upper triangular 3 x 3 square matrix U and a lower triangular 3 x 3 square matrix L.
How To Use the Lu Decomposition Calculator
The user can use the Lu Decomposition Calculator by following the steps given below:
Step 1
The user must first enter the first row of the 3 x 3 square matrix A in the calculator’s input window. The three elements should be entered in curly brackets with commas separating them in the block labeled, “Row 1”.
For the default example, the elements of the first row entered are { 3,1,6 }.
Step 2
The user must now enter the second row of the matrix A in the input tab of the calculator.
To form a square matrix, the user must enter three entries in the block labeled, “Row 2” in between flower brackets with commas separating the elements.
The user enters the second row as { -6,0,-16 } for the default example.
Step 3
The third row of the square matrix A should be entered in the block titled, “Row 3” in the calculator’s input window. For the default example, the entries of the third row are { 0,8,-17 }.
Step 4
The user must now press the “Submit” button for the calculator to process the input 3 x 3 matrix entered by the user.
Output
The calculator displays the output in the following two windows by computing the LU decomposition of the input matrix.
Input
The calculator interprets the input and displays the three input rows in the form of a 3 x 3 square matrix in this output window.
For the default example, the calculator shows the Input interpretation as follows:
\[ LU \ decomposition = \begin{bmatrix} 3 & 1 & 6 \\ -6 & 0 & -16 \\ 0 & 8 & -17 \\ \end{bmatrix} \]
Result
The calculator computes the LU decomposition of the square matrix A by using the equation:
A = LU
For the default example, the calculator displays the A, L, and U as follows:
\[ A = \begin{bmatrix} 3 & 1 & 6 \\ -6 & 0 & -16 \\ 0 & 8 & -17 \\ \end{bmatrix} \]
\[ L = \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 0 & 4 & 1 \\ \end{bmatrix} \]
\[ U = \begin{bmatrix} 3 & 1 & 6 \\ 0 & 2 & -4 \\ 0 & 0 & -1 \\ \end{bmatrix} \]
Solved Example
The following example is solved through the Lu Decomposition Calculator.
Example 1
For the square matrix A given as:
\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & 3 \\ \end{bmatrix} \]
Compute the matrices L and U from the LU decomposition method.
Solution
The user must enter the three rows as { 1,1,1 } , { 4,3, -1 } and { 3,5,3 } in the three input blocks of the calculator.
After submitting the three input rows, the calculator displays the 3 x 3 Input square matrix as follows:
\[ LU \ decomposition = \begin{bmatrix} 1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & 3 \\ \end{bmatrix} \]
The calculator computes the LU decomposition of the input matrix A and displays the three matrices as follows:
\[ A = \begin{bmatrix} 1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & 3 \\ \end{bmatrix} \]
\[ L = \begin{bmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 3 & -2 & 1 \\ \end{bmatrix} \]
\[ U = \begin{bmatrix} 1 & 1 & 1 \\ 0 & -1 & -5 \\ 0 & 0 & -10 \\ \end{bmatrix} \]
