LU Factorization

In Matlab there are several built-in functions provided for matrix factorization (also called decomposition).

The name of the built-in function for a Lower-Upper decomposition is 'lu'. To get the LU factorization of a square matrix A, type the command
'[L, U] = lu(A)'.

Matlab returns a lower triangular matrix L and an upper triangular matrix U such that L*U = A.

Suppose that we need to perform a factorization on

A= [ 1 2 -3
   -3 -4 13
2 1 -5]

We can verify that if matrix

L = [ 1 0 0
-3 1    0
   2 -1.5 1]

and matrix

U = [1 2 -3
   0 2 4
0 0 7]

Then, factors of A are L and U, that is: A = L*U

The decomposition of the matrix A is an illustration of an important and well known theorem. If A is a nonsingular matrix that can be transformed into an upper diagonal form U by the application or row addition operations, then there exists a lower triangular matrix L such that A = LU.

Row addition operations can be represented by a product of elementary matrices. If n such operations are required, the matrix U is related to the matrix A in the following way:

U = E_n E_n-1 ... E₂ E₁ A

The lower triangular matrix L is found from

L = E₁^-1 E₂^-1 ... E_n^-1

L will have ones on the diagonal. The off-diagonal elements are zeros above the diagonal, while the elements below the diagonal are the multipliers required to perform Gaussian elimination on the matrix A. The element l_ij
is equal to the multiplier used to eliminate the (i, j) position.

Example:

In Matlab, let's find the LU decomposition of the matrix

A = [-2 1 -3; 6 -1 8; 8 3 -7]

Write this instruction in the command window or within a script:

[L, U] = lu(A)

And the Matlab answer is:

L =
   -0.2500   -0.5385    1.0000
    0.7500   1.0000           0
    1.0000        0 0

U =
    8.0000    3.0000   -7.0000
         0   -3.2500   13.2500
         0         0   2.3846

We can test the answer, by typing

L*U

And, finnally, the Matlab answer is:

ans =
   -2.0000    1.0000   -3.0000
    6.0000   -1.0000    8.0000
    8.0000    3.0000   -7.0000

>>

Showing that A = L*U, indeed.

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