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QR decomposition. In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares (LLS) problem and is the basis for a particular ...
Householder transformation. In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958 paper by Alston Scott Householder.
Gram–Schmidt process. In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other. By technical definition, it is a method of constructing an orthonormal basis from a set of vectors in an inner ...
Householder's method is a numerical algorithm for solving the equation f(x) = 0. In this case, the function f has to be a function of one real variable. The method consists of a sequence of iterations. beginning with an initial guess x0.
A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size n × n can be constructed as a product of at most n such reflections. A Givens rotation acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosen angle. It is typically used to zero a ...
Givens rotation. In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in the 1950s while he was working at Argonne National Laboratory.
Householder operator. In linear algebra, the Householder operator is defined as follows. [1] Let be a finite-dimensional inner product space with inner product and unit vector . Then. is defined by. {\displaystyle H_ {u} (x)=x-2\,\langle x,u\rangle \,u\,.} This operator reflects the vector across a plane given by the normal vector .
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral ...