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Encyclopedia > LU decomposition

In linear algebra, the LU decomposition is a matrix decomposition which writes a matrix as the product of a lower and upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear equations or calculate the determinant. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ... In linear algebra, a permutation matrix is a binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... A linear equation in algebra is an equation which is constructed by equating two linear functions. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

Contents

Definitions

Let A be a square matrix. An LU decomposition is a decomposition of the form

 A = LU, ,

where L and U are lower and upper triangular matrices (of the same size), respectively. This means that L has only zeros above the diagonal and U has only zeros below the diagonal. For a 3 times 3 matrix, this becomes:

 begin{bmatrix} a_{11} & a_{12} & a_{13}  a_{21} & a_{22} & a_{23}  a_{31} & a_{32} & a_{33}  end{bmatrix} = begin{bmatrix} l_{11} & 0 & 0  l_{21} & l_{22} & 0  l_{31} & l_{32} & l_{33}  end{bmatrix} begin{bmatrix} u_{11} & u_{12} & u_{13}  0 & u_{22} & u_{23}  0 & 0 & u_{33}  end{bmatrix}.

An LDU decomposition is a decomposition of the form

 A = LDU, ,

where D is a diagonal matrix and L and U are unit triangular matrices, meaning that all the entries on the diagonals of L and U are one. In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...


An LUP decomposition is a decomposition of the form

 A = LUP, ,

where L and U are again lower and upper triangular matrices and P is a permutation matrix, i.e., a matrix of zeros and ones that has exactly one entry 1 in each row and column. In linear algebra, a permutation matrix is a binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere. ...


Existence and uniqueness

An invertible matrix admits an LU factorization if and only if all its principal minors are non-zero. The factorization is unique if we require that the diagonal of L (or U) consist of ones. The matrix has a unique LDU factorization under the same conditions. In linear algebra, an n-by-n (square) matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... ↔ ⇔ ≡ logical symbols representing iff. ... In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. ...


If the matrix is singular, then an LU factorization may still exist. In fact, a square matrix of rank k has an LU factorization if the first k principal minors are non-zero. In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...


The exact necessary and sufficient conditions under which a not necessarily invertible matrix over any field has an LU factorization are known. The conditions are expressed in terms of the ranks of certain submatrices. The Gaussian elimination algorithm for obtaining LU decomposition has also been extended to this most general case (Okunev & Johnson 1997).


Every matrix A --square or not-- admits a LUP factorization. The matrices L and P are square matrices, but U has the same shape as A. Upper triangular should be interpreted as having only zero entries below the main diagonal, which starts at the upper left corner. The LUP factorization can be done in such a way that U has only ones on its main diagonal.


Positive definite matrices

If the matrix A is Hermitian and positive definite, then we can arrange matters so that U is the conjugate transpose of L. In this case, we have written A as A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ... In mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...

 A = L L^{*}. ,

This decomposition is called the Cholesky decomposition. The Cholesky decomposition always exists and is unique. Furthermore, computing the Cholesky decomposition is more efficient and numerically more stable than computing the LU decomposition. In mathematics, the Cholesky decomposition, named after André-Louis Cholesky, is a matrix decomposition of a symmetric positive-definite matrix into a lower triangular matrix and the transpose of the lower triangular matrix. ... In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. ...


Explicit Formulation

When an LDU factorization exists and is unique there is a closed (explicit) formula for the elements of L, D, and U in terms of ratios of determinants of certain submatrices of the original matrix A (Householder 1975). In particular, D1 = A1,1 and for i = 2, ldots, n, Di is the ratio of the ith principal submatrix to the (i − 1)th principal submatrix.


Algorithms

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 Doolittle algorithm does the elimination column by column starting from the left, by multiplying A to the left with atomic lower triangular matrices. It results in a unit lower triangular matrix and an upper triangular matrix. The Crout algorithm is slightly different and constructs a lower triangular matrix and a unit upper triangular matrix. In linear algebra, Gaussian elimination is an algorithm that can be used to determine the solutions of a system of linear equations, to find the rank of a matrix, and to calculate the inverse of an invertible square matrix. ...


Computing the LU decomposition using either of these algorithms requires 2n3 / 3 floating point operations, ignoring lower order terms. Partial pivoting adds only a quadratic term and can thus be neglected; this is not the case for full pivoting (Golub & Van Loan 1996). In numerical analysis, pivoting is a process performed on a matrix in order to improve numerical stability, particularly in Gaussian elimination. ...


Doolittle algorithm

Given an N × N matrix

A = (an,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 the 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): = LnA(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 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 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 li,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 − 1A = LU.


LUP algorithm

The LUP decomposition algorithm by Cormen et al. generalizes Crout matrix decomposition. It can be described as follows. In linear algebra, the Crout matrix decomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix (L), an upper triangular matrix (U) and, although not always needed, a permutation matrix (P). ...

  1. If A has a nonzero entry in its first row, then take a permutation matrix P1 such that AP1 has a nonzero entry in its upper left corner. Otherwise, take for P1 the identity matrix. Let A1 = AP1.
  2. Let A2 be the matrix that one gets from A1 by deleting both the first row and the first column. Decompose A2 = L2U2P2 recursively. Make L from L2 by first adding a zero row above and then adding the first column of A1 at the left.
  3. Make U3 from U2 by first adding a zero row above and a zero column at the left and then replacing the upper left entry (which is 0 at this point) by 1. Make P3 from P2 in a similar manner and define A3 = A1 / P3 = AP1 / P3. Let P be the inverse of P1 / P3.
  4. At this point, A3 is the same as LU3, except (possibly) at the first row. If the first row of A is zero, then A3 = LU3, since both have first row zero, and A = LU3P follows, as desired. Otherwise, A3 and LU3 have the same nonzero entry in the upper left corner, and A3 = LU3U1 for some upper triangular square matrix U1 with ones on the diagonal (U1 clears entries of LU3 and adds entries of A3 by way of the upper left corner). Now A = LU3U1P is a decomposition of the desired form.

Small Example

 begin{bmatrix} 4 & 3  6 & 3  end{bmatrix} = begin{bmatrix} l_{11} & 0  l_{21} & l_{22}  end{bmatrix} begin{bmatrix} u_{11} & u_{12}  0 & u_{22}  end{bmatrix}.

One way of finding the LU decomposition of this simple matrix would be to simply solve the linear equations by inspection. You know that:

l11 * u11 + 0 * 0 = 4
l11 * u12 + 0 * u22 = 3
l21 * u11 + l22 * 0 = 6
l21 * u12 + l22 * u22 = 3.

Such a system of equations is underdetermined. In this case any two non-zero elements of L and U matrices are parameters of the solution and can be set arbitrarily to any non-zero value. Therefore to find the unique LU decomposition, it is necessary to put some restriction on L and U matrices. For example, we can require the lower triangular matrix L to be a unit one (i.e. set all the entries of its main diagonal to ones). Then the system of equations has the following solution:

l21 = 1.5
u11 = 4
u12 = 3
u22 = − 1.5.

Substituting these values into the LU decomposition above:

 begin{bmatrix} 4 & 3  6 & 3  end{bmatrix} = begin{bmatrix} 1 & 0  1.5 & 1  end{bmatrix} begin{bmatrix} 4 & 3  0 & -1.5  end{bmatrix}.

Applications

Solving linear equations

Given a matrix equation

A x = L U x = b ,

we want to solve the equation for a given A and b. In this case the solution is done in two logical steps:

  1. First, we solve the equation Ly = b for y
  2. Second, we solve the equation Ux = y for x.

Note that in both cases we have triangular matrices (lower and upper) which can be solved directly using forward and backward substitution without using the Gaussian elimination process (however we need this process or equivalent to compute the LU decomposition itself). Thus the LU decomposition is computationally efficient only when we have to solve a matrix equation multiple times for different b. It is faster to do an LU decomposition of the matrix A once and then solve the triangular matrices for the different b than to use Gaussian elimination each time. In linear algebra, Gaussian elimination is an algorithm that can be used to determine the solutions of a system of linear equations, to find the rank of a matrix, and to calculate the inverse of an invertible square matrix. ...


Inverse matrix

The matrices L and U can be used to calculate the matrix inverse by: In linear algebra, an n-by-n (square) matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

 A^{-1} = U^{-1}L^{-1}. ,

Computer implementations that invert matrices often use this approach.


Determinant

The matrices L and U can be used to compute the determinant of the matrix A very quickly, because det(A) = det(L) det(U) and the determinant of a triangular matrix is simply the product of its diagonal entries. In particular, if L is a unit triangular matrix, then In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

 det(A) = det(L) det(U) = prod_{i=1}^n u_{ii}.

The same approach can be used for LUP decompositions. The determinant of the permutation matrix P is (−1)S, where S is the number of row exchanges in the decomposition.


See also

In linear algebra, a Block LU decomposition is a decomposition of a block matrix into a lower block triangular matrix L and an upper block triangular matrix U. This decomposition is used in numerical analysis to reduce the complexity of the block matrix formula. ... In mathematics, the Cholesky decomposition, named after André-Louis Cholesky, is a matrix decomposition of a symmetric positive-definite matrix into a lower triangular matrix and the transpose of the lower triangular matrix. ... In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. ...

References

  • Cormen, T.H.; Leisserson, C.E & Rivest, R.L., Introduction to Algorithms 
  • Golub, Gene H. & Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins, ISBN 978-0-8018-5414-9 .
  • Horn, Roger A. & Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 0-521-38632-2 . See Section 3.5.
  • Okunev, Pavel & Johnson, Charles (1997), Necessary And Sufficient Conditions For Existence of the LU Factorization of an Arbitrary Matrix, arXiv:math.NA/0506382 .
  • Householder, Alston (1975), The Theory of Matrices in Numerical Analysis .
  • LU decomposition on MathWorld.
  • LU decomposition on Math-Linux.

Gene Golub in 2007 Gene Howard Golub (February 29, 1932 – November 16, 2007), Fletcher Jones Professor of Computer Science (and, by courtesy, of Electrical Engineering) at Stanford University, was one of the preeminent numerical analysts of his generation. ... Professor Charles Francis Van Loan, the current chair of the Department of Computer Science at Cornell University, is known for his enthusiasm for teaching and for his expertise in numerical analysis, especially matrix computations. ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ...

External links

Ed Pegg, Jr. ...

  Results from FactBites:
 
PlanetMath: LU decomposition (305 words)
The LU factorization is closely related to the row reduction algorithm.
The key idea behind LU factorization is that one does not need to employ row scalings to do row reduction until the second half (the back-substitution phase) of the algorithm.
This is version 6 of LU decomposition, born on 2002-01-04, modified 2006-09-11.
PlanetMath: LU decomposition (305 words)
The LU factorization is closely related to the row reduction algorithm.
The key idea behind LU factorization is that one does not need to employ row scalings to do row reduction until the second half (the back-substitution phase) of the algorithm.
This is version 6 of LU decomposition, born on 2002-01-04, modified 2006-09-11.
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