The formula to calculate the LU decomposition of a matrix is:
\[ A = LU \]
Where:
LU Decomposition, also known as LU factorization, is a method in numerical analysis for solving linear equations, inverting matrices, and computing determinants. The term "LU" stands for "Lower Upper", and it decomposes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The lower triangular matrix has ones on the diagonal and the elements above the diagonal are zero, while the upper triangular matrix has zeros below the diagonal. This decomposition is mainly used to simplify the solution of a system of linear equations, as it can be applied to the system's matrix. LU Decomposition is particularly useful when the system needs to be solved repeatedly for different right-hand sides, as it reduces the computational cost.
Example:
Step 1: Calculate the LU decomposition:
Lower Triangular Matrix (L):
\[
\begin{pmatrix}
1 & 0 \\
1.5 & 1
\end{pmatrix}
\]
Upper Triangular Matrix (U):
\[
\begin{pmatrix}
4 & 3 \\
0 & -1.5
\end{pmatrix}
\]
