What is L U Decomposition?

A square matrix A can be decomposed into two square matrices L and U such that A = L U where U is an upper triangular matrix formed as a result of applying the Gauss Elimination Method on A, and L is a lower triangular matrix with diagonal elements being equal to 1. 

For A  = [Tex]\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} [/Tex].

We have L =[Tex] \begin{bmatrix} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{bmatrix} [/Tex] and U = [Tex]\begin{bmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{bmatrix} [/Tex];

Such that A = L U i.e., [Tex]\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right]=\left[\begin{array}{lll} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 0 \end{array}\right] \cdot \left[\begin{array}{ccc} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{array}\right][/Tex]

Here the value of l₂₁, u₁₁, etc. can be compared and found.

What is Gauss Elimination Method?

Gaussian Elimination, also known as Gauss-Jordan Elimination, is a method used in linear algebra to solve systems of linear equations and to find the inverse of a matrix. It’s named after the mathematician Carl Friedrich Gauss and also the mathematician Wilhelm Jordan, who made significant contributions to its development.

According to the Gauss elimination method:

1. Any zero row should be at the bottom of the matrix.
2. The first non-zero entry of each row should be on the right-hand side of the first non-zero entry of the preceding row. This method reduces the matrix to row echelon form.

LU decomposition of a matrix is the factorization of a given square matrix into two triangular matrices, one upper triangular matrix and one lower triangular matrix, such that the product of these two matrices gives the original matrix. It was introduced by Alan Turing in 1948, who also created the Turing machine.

LU decomposition method of factorizing a matrix as a product of two triangular matrices has various applications such as a solution of a system of equations, which itself is an integral part of many applications such as finding current in a circuit and solution of discrete dynamical system problems; finding the inverse of a matrix and finding the determinant of the matrix.