LU decomposition
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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.
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Definition
Let A be an invertible matrix over a field. The matrix A can be decomposed as
- <math>P^{-1}A = L U <math>
- <math>A = L UP <math>
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 <math>A = L U.<math>
Notes
If the matrix A is positive definite we can construct the even simpler Cholesky decomposition
- <math>A = L L^{*}<math>
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
- <math>
A= (a_{n,n}) <math> we define
- <math> A^{(0)} := A<math>
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
- <math>l_{i,n} := -\frac{a_{i,n}^{(n-1)}}{a_{n,n}^{(n-1)}}<math>
for <math>i = n+1,\ldots,N<math>. This can be done by multiplying A(n-1) to the left with the lower triangular matrix
- <math>
L_n = \begin{pmatrix}
1 & & & & & 0 \\
& \ddots & & & & \\
& & 1 & & & \\
& & l_{n+1,n} & \ddots & & \\
& & \vdots & & \ddots & \\
0 & & l_{N,n} & & & 1 \\
\end{pmatrix}. <math>
We set
- <math> A^{(n)} := L_n A^{(n-1)}.<math>
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
- <math>
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)}. <math>
Denote the upper triangular matrix A(N-1) by U, and <math>L=L_{1}^{-1} \ldots L_{N-1}^{-1}<math>. Because the inverse of a lower triangular matrix Ln 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 <math>A=LU<math>.
It is clear that in order for this algorithm to work, one needs to have <math>a_{n,n}^{(n-1)}\not=0<math> at each step (see the definition of <math>l_{i,n}<math>). 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 <math>P^{-1}A = L U <math>.
Crout algorithm
Main article Crout matrix decomposition
Applications
Solving linear equations
Given a matrix equation
- <math>A x = L U x = b<math>
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.
Related articles
See also
it:Decomposizione LU ja:LU分解 nl:LU-decompositie pl:Metoda LU