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The first two steps of the 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.
The Gram matrix is symmetric in the case the inner product is real-valued; it is Hermitian in the general, complex case by definition of an inner product. The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can ...
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.
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).
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
Thus the problem of finding conjugate axes is less constrained than the problem of orthogonalization, so the Gram–Schmidt process works, with additional degrees of freedom that we can later use to pick the ones that would simplify the computation: Arbitrarily set .
We could use an orthogonal projection (Gram-Schmidt) but this will be numerically unstable if the vectors and are close to orthogonal. Instead, the Householder reflection reflects through the dotted line (chosen to bisect the angle between x {\displaystyle \mathbf {x} } and e 1 {\displaystyle \mathbf {e} _{1}} ).
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 .
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