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  2. Gram–Schmidt process - Wikipedia

    en.wikipedia.org/wiki/GramSchmidt_process

    The calculation of the sequence , …, is known as Gram–Schmidt orthogonalization, and the calculation of the sequence , …, is known as Gram–Schmidt orthonormalization. To check that these formulas yield an orthogonal sequence, first compute u 1 , u 2 {\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle } by substituting the ...

  3. QR decomposition - Wikipedia

    en.wikipedia.org/wiki/QR_decomposition

    This method has greater numerical stability than the Gram–Schmidt method above. 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.

  4. Iwasawa decomposition - Wikipedia

    en.wikipedia.org/wiki/Iwasawa_decomposition

    In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of Gram–Schmidt orthogonalization).

  5. Vector projection - Wikipedia

    en.wikipedia.org/wiki/Vector_projection

    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.

  6. Schmidt decomposition - Wikipedia

    en.wikipedia.org/wiki/Schmidt_decomposition

    In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has numerous applications in quantum information theory , for example in entanglement characterization and in state purification , and plasticity .

  7. Orthonormality - Wikipedia

    en.wikipedia.org/wiki/Orthonormality

    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.

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