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In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v1, ... , vk} in an inner product space (most commonly the Euclidean space Rn), orthogonalization results in a set of orthogonal vectors {u1, ... , uk} that generate the same subspace as the vectors v1, ... , vk. Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span.
In addition, if we want the resulting vectors to all be unit vectors, then we normalize each vector and the procedure is called orthonormalization.
Orthogonalization is also possible with respect to any symmetric bilinear form (not necessarily an inner product, not necessarily over real numbers), but standard algorithms may encounter division by zero in this more general setting.
Givens rotation Symmetric orthogonalization, which uses the Singular value decomposition When performing orthogonalization on a computer, the Householder...
is Gram–Schmidt orthogonalization of columns of A, started from the first column. RQ decomposition is Gram–Schmidt orthogonalization of rows of A, started...
triangular matrix (QR decomposition, a consequence of Gram–Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who...
signals to a greater or lesser degree. The local signal-and-noise orthogonalization algorithm can be used to avoid changes to the signals. Boosting signals...
can extract an orthogonal basis, via a method such as Gram–Schmidt orthogonalization. The resulting set of vectors is thus an orthogonal basis of the Krylov...
gradually lost its true orthogonality. A Gram–Schmidt process could orthogonalize the columns, but it is not the most reliable, nor the most efficient...
between HL and memory was now possible, using a direct address. The Z80 orthogonalized this further by making all 16-bit register pairs, including IX and IY...
GSL also offers an alternative method that uses a one-sided Jacobi orthogonalization in step 2 (GSL Team 2007). This method computes the SVD of the bidiagonal...
set dense in C[−π,π] and therefore an orthonormal basis of C[−π,π]. Orthogonalization Orthonormal function system Axler, Sheldon (1997), Linear Algebra...
Geosynchronous orbit, an orbit around Earth of a satellite Gram-Schmidt orthogonalization, in mathematics GSO projection, in superstring theory General Staff...
with respect to which the following methods differ, is whether the orthogonalization of the basis functionals is to be performed over the idealized specification...
iteration, methods relying on Krylov subspace frequently involve some orthogonalization scheme, such as Lanczos iteration for Hermitian matrices or Arnoldi...
There are other methods than the Cholesky decomposition in use. Orthogonalization methods (such as QR factorization) are common, for example, when solving...
an integer set of coefficients or prefers tighter formulas if the orthogonalization is involved.) The explicit representation is R n ( l ) ( ρ ) = 2 n...
iteration and matrix deflation by subtraction. A Gram–Schmidt re-orthogonalization algorithm is applied to both the scores and the loadings at each iteration...
multiple source mixture signals, we can use kurtosis and Gram-Schmidt Orthogonalization (GSO) to recover the signals. Given M signal mixtures in an M-dimensional...
linear forms Dual pair Orthogonality – Various meanings of the terms Orthogonalization Bhushan, Datta, Kanti (2008). Matrix And Linear Algebra, Edition 2:...
complete basis of vectors, using QR decomposition to renormalize (and orthogonalize). For a symmetric matrix A, upon convergence, AQ = QΛ, where Λ is the...
are constructed from the derivatives of γ(t) using the Gram–Schmidt orthogonalization algorithm with e 1 ( t ) = γ ′ ( t ) ‖ γ ′ ( t ) ‖ e j ( t ) = e j...
Min-max theorem Normal vector Orthonormal basis Orthogonal complement Orthogonalization Parallelogram law Normal matrix, normal operator Orthogonal matrix...
auxiliary observations in yk. The l·d·lt square-root filter requires orthogonalization of the observation vector. This may be done with the inverse square-root...
to construct orthonormal bases, including using the Gram–Schmidt orthogonalization or singular-value decomposition of clr transformed data. Another alternative...
biological systems Modified Gram-Schmidt, a mathematical method for orthogonalizing a set of vectors; see Gram–Schmidt process § Numerical stability Metal...
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