LU decomposition
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LU decomposition, or Doolittle decomposition is the process of decomposing a matrix into a product of an upper-triangular matrix U and a lower triangular matrix L. LU decomposition is LUP decomposition, where P is the identity matrix. In this article, it is represented by the variable T.

Factorization

The basic line of thought to get this upper and lower triangular matrix is as follows. Using gaussian elimination on a matrix A we actually apply a number of elementary matrix transformations to obtain an upper triangular matrix U and then use back substitution to quickly solve the problem.

Suppose T is the product of all elementary transformations and applying it to A, this gives us:

TAx=Tb, with TA=U, upper triangular,

resulting in Ux=Tb.

The gaussian elimination algorithm uses only the third form of the three elementary matrix transformations to make TA upper triangular. Using the properties of these elementary transformations, we calculate the inverse of T which shows us it is a lower triangular matrix L which can be added to the left and right side of the previous equation resulting in:

Ux=T₁ T₂ ... T_n b, with T₁^-1 T₂^-1 ... T_n^-1 = L

⇒ LUx=b.

Solving

Once the upper and lower triangular matrices are known, solving this equation becomes very efficient as it can be split up into:

Lz=b using forward substitution

and

Ux=x using backward substition, both solvable in O(n²) time.

Another method

Here is a direct method for decomposing an matrix into the product of a lower triangular matrix and an upper triangular matrix ,

For a matrix, this becomes:

This gives three types of equations

This gives equations for unknowns (the decomposition is therefore not unique); the system can be solved using Crout's method. To solve the matrix equation

first solve for . This can be done by forward substitution

for i=2,...,N. Then solve for . This can be done by back substitution

for i=N-1,...,1

Applications

The matrices L and U can be used to calculate the matrix inverse. Computer implementations that invert matrices often use this approach.

Related articles

• Numerical analysis
• Cholesky decomposition
• Crout decomposition

The original article can be found at: http://en.wikipedia.org/wiki/LU_decomposition.
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