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The algorithms using Householder transformations are more stable than the stabilized Gram–Schmidt process. On the other hand, the Gram–Schmidt process produces the th orthogonalized vector after the th iteration, while orthogonalization using Householder reflections produces all the vectors only at the end. This makes only the Gram ...
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: A P = Q R A = Q R P T {\displaystyle AP=QR\quad \iff \quad A=QRP^{\textsf {T}}}
The vector projection is an important operation in the Gram–Schmidt orthonormalization of vector space bases. It is also used in the separating axis theorem to detect whether two convex shapes intersect.
Magma as the functions LLL and LLLGram (taking a gram matrix) Maple as the function IntegerRelations[LLL] Mathematica as the function LatticeReduce; Number Theory Library (NTL) as the function LLL; PARI/GP as the function qflll; Pymatgen as the function analysis.get_lll_reduced_lattice; SageMath as the method LLL driven by fpLLL and NTL
On the other hand, the Gram–Schmidt process produces the jth orthogonalized vector after the jth iteration, while orthogonalization using Householder reflections produces all the vectors only at the end. This makes only the Gram–Schmidt process applicable for iterative methods like the Arnoldi iteration.
In other words, the sequence is obtained from the sequence of monomials 1, x, x 2, … by the Gram–Schmidt process with respect to this inner product. Usually the sequence is required to be orthonormal , namely, P n , P n = 1 , {\displaystyle \langle P_{n},P_{n}\rangle =1,} however, other normalisations are sometimes used.
The Gram-Schmidt theorem, together with the axiom of choice, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors.
Bra–ket notation; Definite bilinear form; Direct integral; Euclidean space; Fundamental theorem of Hilbert spaces; Gram–Schmidt process; Hellinger–Toeplitz theorem