QR decomposition

If A is invertible, then the factorization is unique if we require the diagonal elements of R to be positive.[1] If A is of full rank n and we require that the diagonal elements of R1 are positive then R1 and Q1 are unique, but in general Q2 is not.R1 is then equal to the upper triangular factor of the Cholesky decomposition of A* A (= ATA if A is real).Analogously, we can define QL, RQ, and LQ decompositions, with L being a lower triangular matrix.There are several methods for actually computing the QR decomposition, such as the Gram–Schmidt process, Householder transformations, or Givens rotations.If the algorithm is implemented using floating-point arithmetic, then α should get the opposite sign as the k-th coordinate ofis to be the pivot coordinate after which all entries are 0 in matrix A's final upper triangular form, to avoid loss of significance.is an m-by-m Householder matrix, which is both symmetric and orthogonal (Hermitian and unitary in the complex case), and This can be used to gradually transform an m-by-n matrix A to upper triangular form.This results in a matrix Q1A with zeros in the left column (except for the first row).This can be repeated for A′ (obtained from Q1A by deleting the first row and first column), resulting in a Householder matrix Q′2.Since we want it really to operate on Q1A instead of A′ we need to expand it to the upper left, filling in a 1, or in general: AfterThe following table gives the number of operations in the k-th step of the QR-decomposition by the Householder transformation, assuming a square matrix with size n. Summing these numbers over the n − 1 steps (for a square matrix of size n), the complexity of the algorithm (in terms of floating point multiplications) is given by Let us calculate the decomposition of First, we need to find a reflection that transforms the first column of matrix A, vectorTake the (1, 1) minor, and then apply the process again to By the same method as above, we obtain the matrix of the Householder transformation after performing a direct sum with 1 to make sure the next step in the process works properly.The use of Householder transformations is inherently the most simple of the numerically stable QR decomposition algorithms due to the use of reflections as the mechanism for producing zeroes in the R matrix.This algorithm can be applied in the case when the matrix A has m >> n.[3] This algorithm uses a binary reduction tree to compute local householder QR decomposition at each node in the forward pass, and re-constitute the Q matrix in the backward pass.The binary tree structure aims at decreasing the amount of communication between processor to increase performance.The concatenation of all the Givens rotations forms the orthogonal Q matrix.The Givens rotation procedure is useful in situations where only relatively few off-diagonal elements need to be zeroed, and is more easily parallelized than Householder transformations.Let us calculate the decomposition of First, we need to form a rotation matrix that will zero the lowermost left element,The QR decomposition via Givens rotations is the most involved to implement, as the ordering of the rows required to fully exploit the algorithm is not trivial to determine.This makes the Givens rotation algorithm more bandwidth efficient and parallelizable than the Householder reflection technique.by introducing the definition of QR decomposition for non-square complex matrices and replacing eigenvalues with singular values.From the properties of the singular value decomposition (SVD) and the determinant of a matrix, we have where theHowever, if A is square, then It follows that the QR decomposition can be used to efficiently calculate the product of the eigenvalues or singular values of a matrix.Pivoted QR differs from ordinary Gram-Schmidt in that it takes the largest remaining column at the beginning of each new step—column pivoting—[4] and thus introduces a permutation matrix P: Column pivoting is useful when A is (nearly) rank deficient, or is suspected of being so.This can be used to find the (numerical) rank of A at lower computational cost than a singular value decomposition, forming the basis of so-called rank-revealing QR algorithms.After some algebra, it can be shown that a solution to the inverse problem can be expressed as:The latter technique enjoys greater numerical accuracy and lower computations.Equivalent to the underdetermined case, back substitution can be used to quickly and accurately find thisare often provided by numerical libraries as an "economic" QR decomposition.)