lu decomposition

In linear algebra, a LU decomposition, or LUP decomposition is a matrix decomposition of a matrix into a lower triangular matrix L, an upper-triangular matrix U, and a permutation matrix P. This decomposition is used in numerical analysis to solve systems of linear equations.

Definition

Let A be an invertible matrix over a field. The matrix A can be decomposed as

P^{-1}A = L U

A = L UP

where P is a permutation matrix (as is P^-1), L is a lower triangular matrix, and U is an upper triangular matrix. Sometimes the permutation matrix can be chosen to be the identity matrix. In this case the decomposition becomes

A = L U.

Notes

If the matrix A is positive definite we can construct the even simpler Cholesky decomposition

A = L L^{*}

with L a lower triangular matrix with positive diagonal entries, and L^* the conjugate transpose of L.

Main idea

The LU decomposition is basically a modified form of Gaussian elimination. We transform the matrix A into an upper triangular matrix U by eliminating the entries below the main diagonal. The elimination is done column by column starting from the left, by multiplying A to the left with atomic lower triangular matrices.

Algorithms

There are two different algorithms to construct a LU-decomposition. The Doolittle decomposition constructs a unit lower triangular matrix and an upper triangular matrix, whereas the Crout algorithm constructs a lower triangular matrix and an unit upper triangular matrix.

Doolittle algorithm

Given an NxN matrix

A= (a_{n,n}) we define

A^{(0)} := A

and then we iterate n = 1,...,N-1 as follows. We eliminate the matrix elements below the main diagonal in the n-th column of A^(n-1) by adding to i-th row of this matrix the n-th row multiplied by

l_{i,n} := -\frac{a_{i,n}^{(n-1)}}{a_{n,n}^{(n-1)}}

for

i = n+1,\ldots,N

. This can be done by multiplying A^(n-1) to the left with the lower triangular matrix

L_n = \begin{pmatrix}

      1 &        &           &         &         & 0 \\        & \ddots &           &         &         &   \\        &        &         1 &         &         &   \\        &        & l_{n+1,n} &  \ddots &         &   \\        &        &    \vdots &         &  \ddots &   \\      0 &        &   l_{N,n} &         &         & 1 \\

\end{pmatrix}. We set

A^{(n)} := L_n A^{(n-1)}.

After N-1 steps, we eliminated all the matrix elements below the main diagonal, so we obtain an upper triangular matrix A^(N-1). We find the decomposition

A = L_{1}^{-1} L_{1} A^{(0)} = L_{1}^{-1} A^{(1)} = L_{1}^{-1} L_{2}^{-1} L_{2} A^{(1)} = L_{1}^{-1}L_{2}^{-1} A^{(2)} =\ldots = L_{1}^{-1} \ldots L_{N-1}^{-1} A^{(N-1)}. Denote the upper triangular matrix A^(N-1) by U, and

L=L_{1}^{-1} \ldots L_{N-1}^{-1}

. Because the inverse of a lower triangular matrix L_n is again a lower triangular matrix, and the multiplication of two lower triangular matrices is again a lower triangular matrix, it follows that L is a lower triangular matrix. We obtain

A=LU

. It is clear that in order for this algorithm to work, one needs to have

a_{n,n}^{(n-1)}\not=0

at each step (see the definition of

l_{i,n}

). If this assumption fails at some point, one needs to interchange n-th row with another row below it before continuing. This is why the LU decomposition in general looks like

P^{-1}A = L U

Crout algorithm

Main article Crout matrix decomposition

Applications

Solving linear equations

Given a matrix equation

A x = L U x = b

we want to solve the matrix A for a given b. Triangular matrices (upper and lower) can be solved directly using forward and backward substitution without using the slow Gaussian elimination process. So when we have to solve a matrix equation multiple times for different b it is faster to do a LU-decomposition of the matrix once and then solve the triangular matrices for the different b than to use Gaussian elimination each time.

Inverse matrix

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