How to Solve LU Decomposition / Factorization - Tutorial

How to Solve LU Decomposition / Factorization Matrix - Definition, Formula, Example

Definition:

Lu ( 'Lower Upper') decomposition is one which factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. In lower triangle matrix, the diagonal is one, and upper part of the diagonal is zero. In upper triangle matrix, the lower part of diagonal is zero. It is also called as LU Factorization of Matrix.

Formula:

A = PLU Where A = Square Matrix P = Permutation Matrix L = Lower Triangular Matrix U = Upper Triangular Matrix

Example :

Consider the Square Matrix (A) = 1 2 -3
-8 5 2
6 -4 0

Given :

Square Matrix (A) = (A11) 1 (A12) 2 (A13) -3
(A21) -8 (A22) 5 (A23) 2
(A31) 6 (A32) -4 (A33) 0

To Find,

Lower and Upper Triangular Matrix

Solution :

Step 1:

The given Square Matrix order i.e., n = 3

Permutation Matrix (P) = 0 0 1
0 1 0
1 0 0
Step 2:

Now, let us form the lower and upper triangular matrix.

In lower triangle matrix (L), the diagonal is one, and upper part of the diagonal is zero.

So, Lower Triangular (L) = 1 0 0
a21 1 0
a31 a32 1

In upper triangle matrix, the lower part of diagonal is zero.

Upper Triangular (U) = b11 b12 b13
0 b22 b23
0 0 b33
Step 3:

In upper triangular matrix,

uij = aij - (k=1, Σ, i-1) ukj lik

Apply the above condition in upper triangular matrix,

Upper Triangular (U) = 6 -4 0
0 -0.333 2
0 0 13
Step 4:

In lower triangular matrix,

lij =(1/uij) (aij - (k=1, Σ, j-1) ukj lik)

Apply the above condition in lower triangular matrix

Lower Triangular (L) = 1 0 0
-1.333 1 0
0.167 -8 1

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