Expanding on what J W linked, let the matrix be positive definite be such that it can be represented as a Cholesky decomposition, A = L L − 1. Defines LDU factorization. Illustrates the technique using Tinney’s method of LDU decomposition. Recall from The LU Decomposition of a Matrix page that if we have an matrix We will now look at some concrete examples of finding an decomposition of a.
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In numerical analysis and linear algebralower—upper LU decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well.
LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. LU decomposition was introduced by mathematician Tadeusz Banachiewicz in Let A be a square matrix. In the lower triangular matrix all elements above the diagonal are zero, in the upper triangular matrix, all the elements below the diagonal are zero.
Without a proper ordering or permutations in the matrix, the factorization may fail to materialize. This is impossible if A is nonsingular invertible. This is a procedural secomposition.
It can be removed by simply reordering the rows of A so that the first element of the permuted matrix is nonzero. The same problem in subsequent factorization steps can be removed the same way; see the basic procedure below. It turns out that deomposition proper permutation in rows or columns is sufficient for LU factorization. It turns out that all square matrices can be factorized in this form,  and the factorization is numerically stable in practice.
Linear Algebra, Part 8: A=LDU Matrix Factorization
Above we required that A be a square matrix, but these decompositions can all be generalized to rectangular matrices as well. In that case, L and D are square matrices both of which have the same number of rows as Aand U has exactly the same dimensions as A. Upper triangular should be interpreted as having only zero entries below the main diagonal, which starts at the upper left corner. One way to find the LU decomposition of this simple matrix would be to simply solve the linear equations by inspection.
Expanding the matrix multiplication gives. This 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 conveniently require the lower triangular matrix L to be a unit triangular matrix i. Then the system of equations has the following solution:. If a square, invertible matrix has an LDU factorization with all diagonal entries of L and U equal to 1, then the factorization is unique. If A is a symmetric or Hermitianif A is complex positive definite matrix, we can arrange matters so that U is the conjugate transpose of L. That is, we can write A as. This decomposition is called the Cholesky decomposition.
The Cholesky decomposition always exists and is unique — provided the matrix is positive definite. Furthermore, computing the Cholesky decomposition is more efficient and numerically more stable than computing some other LU decompositions.
For a not necessarily invertible matrix over any field, the exact necessary and sufficient conditions under which it 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. 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. Partial pivoting adds only a quadratic term; this is not the case for full pivoting.
When an LDU factorization exists and is unique, there is a closed explicit formula for the elements of LDand U in terms of ratios of determinants of certain submatrices of the original matrix A.
Computation of the determinants is computationally expensiveso this explicit formula is not used in practice. We find the decomposition. 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.
Moreover, it can be seen that. If this assumption fails at some point, one needs to interchange n -th row with another row below it before continuing. The LUP decomposition algorithm by Cormen et al. It can be described as follows.
It is possible to find a low rank approximation to an LU decomposition using a randomized algorithm. Special algorithms have been developed for factorizing large sparse matrices. These algorithms attempt to find sparse factors L and U. Ideally, the cost of computation is determined by the number of nonzero entries, rather than by the size of the matrix. These algorithms use the freedom to exchange rows and columns to minimize fill-in entries that change from an initial zero to a non-zero value during the execution of an algorithm.
General treatment of orderings that minimize fill-in can be addressed using graph theory. Note that in both cases we are dealing with triangular matrices L and Uwhich can be solved directly by forward and backward substitution without using the Gaussian elimination process however we do need this process or equivalent to compute the LU decomposition itself.
The above procedure can be repeatedly applied to solve the equation multiple times for different b. In this case it is faster and more convenient to do an LU decomposition of the matrix A once and then solve the triangular matrices for the different brather than using Gaussian elimination each time.
The matrices L and U could be thought to have “encoded” the Gaussian elimination process. For this reason, LU decomposition is usually preferred.
When solving systems of equations, b is usually treated as a vector with a length equal to the height of matrix A. In matrix inversion however, instead of vector bwe have matrix Bwhere B is an n -by- p matrix, so that we are trying to find a matrix X also a n -by- p matrix:.
We can use the same algorithm presented earlier to solve for each column of matrix X. Now suppose that B is the identity matrix of size n.
Linear Algebra, Part 8: A=LDU Matrix Factorization – Derivative Works
It would follow that the result X must be the inverse of A. The same method readily applies to LU decomposition by setting P equal to the identity matrix. From Wikipedia, the free encyclopedia. Astronomy and Astrophysics Supplement. Applied and Computational Harmonic Analysis.