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If A is invertible, then the factorization is unique if we require the diagonal elements of R to be positive. If A has n linearly independent columns, then the first n columns of Q form an orthonormal basis for the column space of A. Q1 and Q2 both have orthogonal columns. Trefethen and Bau call this the reduced QR factorization. Analogously, we can define QL, RQ, and LQ decompositions, with L being a lower triangular matrix.

Schmidt process, Householder transformations, or Givens rotations. Each has a number of advantages and disadvantages. The only difference from QR decomposition is the order of these matrices. Schmidt orthogonalization of columns of A, started from the first column.

Schmidt orthogonalization of rows of A, started from the last row. The Gram-Schmidt process is inherently numerically unstable. While the application of the projections has an appealing geometric analogy to orthogonalization, the orthogonalization itself is prone to numerical error. A significant advantage however is the ease of implementation, which makes this a useful algorithm to use for prototyping if a pre-built linear algebra library is unavailable. The maximum angle with this transform is 45 degrees.

Q can be used to reflect a vector in such a way that all coordinates but one disappear. 0 in matrix A’s final upper triangular form, to avoid loss of significance. This can be used to gradually transform an m-by-n matrix A to upper triangular form. First, we multiply A with the Householder matrix Q1 we obtain when we choose the first matrix column for x.

The 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. 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. However, the Householder reflection algorithm is bandwidth heavy and not parallelisable, as every reflection that produces a new zero element changes the entirety of both Q and R matrices. QR decompositions can also be computed with a series of Givens rotations. Each rotation zeroes an element in the subdiagonal of the matrix, forming the R matrix. The concatenation of all the Givens rotations forms the orthogonal Q matrix.

In practice, Givens rotations are not actually performed by building a whole matrix and doing a matrix multiplication. A Givens rotation procedure is used instead which does the equivalent of the sparse Givens matrix multiplication, without the extra work of handling the sparse elements. It describes the layout and organization of the Unicode Character Database and how it specifies the formal definitions of the Unicode Character Properties. Status This document has been reviewed by Unicode members and other interested parties, and has been approved for publication by the Unicode Consortium. This is a stable document and may be used as reference material or cited as a normative reference by other specifications.

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