#### 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,

u_{ij} = a_{ij} - (k=1, Σ, i-1) u_{kj} l_{ik}

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,

l_{ij} =(1/u_{ij}) (a_{ij} - (k=1, Σ, j-1) u_{kj} l_{ik})

Apply the above condition in lower triangular matrix

Lower Triangular (L) = | 1 | 0 | 0 |

-1.333 | 1 | 0 |

0.167 | -8 | 1 |