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 nÃ—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. ...
## Definitions
Let *A* be a square matrix. An **LU decomposition** is a decomposition of the form 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 matrix, this becomes: An **LDU decomposition** is a decomposition of the form 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 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. ...
â†” â‡” â‰¡ 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 â€” 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. ...
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 AndrÃ©-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, *D*_{1} = *A*_{1,1} and for , *D*_{i} is the ratio of the *i*^{th} 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 2*n*^{3} / 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* = (*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 the *i*-th row of this matrix the *n*-th row multiplied by for . This can be done by multiplying *A*^{(n-1)} to the left with the lower triangular matrix 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 Denote the upper triangular matrix *A*^{(N-1)} by *U*, and . 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* = *L**U*. It is clear that in order for this algorithm to work, one needs to have 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*.
### 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). ...
- If
*A* has a nonzero entry in its first row, then take a permutation matrix *P*_{1} such that *A**P*_{1} has a nonzero entry in its upper left corner. Otherwise, take for *P*_{1} the identity matrix. Let *A*_{1} = *A**P*_{1}. - Let
*A*_{2} be the matrix that one gets from *A*_{1} by deleting both the first row and the first column. Decompose *A*_{2} = *L*_{2}*U*_{2}*P*_{2} recursively. Make *L* from *L*_{2} by first adding a zero row above and then adding the first column of *A*_{1} at the left. - Make
*U*_{3} from *U*_{2} 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 *P*_{3} from *P*_{2} in a similar manner and define *A*_{3} = *A*_{1} / *P*_{3} = *A**P*_{1} / *P*_{3}. Let *P* be the inverse of *P*_{1} / *P*_{3}. - At this point,
*A*_{3} is the same as *L**U*_{3}, except (possibly) at the first row. If the first row of *A* is zero, then *A*_{3} = *L**U*_{3}, since both have first row zero, and *A* = *L**U*_{3}*P* follows, as desired. Otherwise, *A*_{3} and *L**U*_{3} have the same nonzero entry in the upper left corner, and *A*_{3} = *L**U*_{3}*U*_{1} for some upper triangular square matrix *U*_{1} with ones on the diagonal (*U*_{1} clears entries of *L**U*_{3} and adds entries of *A*_{3} by way of the upper left corner). Now *A* = *L**U*_{3}*U*_{1}*P* is a decomposition of the desired form. ## Small Example One way of finding the LU decomposition of this simple matrix would be to simply solve the linear equations by inspection. You know that: *l*_{11} * *u*_{11} + 0 * 0 = 4 *l*_{11} * *u*_{12} + 0 * *u*_{22} = 3 *l*_{21} * *u*_{11} + *l*_{22} * 0 = 6 *l*_{21} * *u*_{12} + *l*_{22} * *u*_{22} = 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: *l*_{21} = 1.5 *u*_{11} = 4 *u*_{12} = 3 *u*_{22} = − 1.5. Substituting these values into the LU decomposition above: ## Applications ### Solving linear equations Given a matrix equation we want to solve the equation for a given *A* and *b*. In this case the solution is done in two logical steps: - First, we solve the equation
*L**y* = *b* for *y* - Second, we solve the equation
*U**x* = *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. ...
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 nÃ—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. ...
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 AndrÃ©-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 â€“ 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 Ï‡) 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. ...
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