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# LU Decomposition Definition/Meaning:

A method used in numerical linear algebra in order to solve a set of linear equations,

Ax = b

where A is a square matrix and b is a column vector. In this method, a lower triangular matrix L and an upper triangular matrix U are sought such that

LU = A

For definiteness, the diagonal elements of L may be taken to be 1. The elements of successive rows of U and L may easily be calculated from the defining equations. Once L and U have been determined, so that

LUx = b,

the equation

Ly = b

is found by forward substitution.

Thereafter the equation

Ux = y

is found by backward substitution, x is then the solution to the original problem. A variant of the method, the method of LDU decomposition, seeks lower and upper triangular matrices with unit diagonal and a diagonal matrix D, such that

A = LDU

If the matrix A is symmetric and positive definite, there is an advantage in finding a lower triangular matrix L such that

A = LLT

This method is known as Cholesky decomposition; the diagonal elements of L are not, in general, unity.

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