Schmidt orthogonalization

E384564

algorithm linear algebra concept mathematical procedure

Schmidt orthogonalization is a mathematical procedure, also known as the Gram–Schmidt process, that converts a set of linearly independent vectors into an orthonormal set spanning the same subspace.

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All labels observed (6)

Label	Occurrences
Gram–Schmidt orthogonalization	1
Gram–Schmidt orthonormalization	1
Gram–Schmidt process	1
Schmidt orthogonalization canonical	1
block Gram–Schmidt process	1
modified Gram–Schmidt process	1

How this entity was disambiguated

This entity first appeared as the object of triple T3748401 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Schmidt orthogonalization
Context triple: [Erhard Schmidt, notableWork, Schmidt orthogonalization]

A. Richardson iteration
Richardson iteration is an early iterative method for solving linear systems and other operator equations, based on repeated relaxation steps to progressively improve an approximate solution.
B. Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
C. Gaussian elimination
Gaussian elimination is a fundamental algorithm in linear algebra used to solve systems of linear equations by systematically transforming matrices into row-echelon form.
D. Jacobi method
The Jacobi method is an iterative numerical algorithm used to solve systems of linear equations by repeatedly updating each variable using values from the previous iteration.
E. Gauss–Seidel method
The Gauss–Seidel method is an iterative numerical technique used to solve systems of linear equations, particularly in large, sparse problems arising in scientific and engineering computations.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Schmidt orthogonalization
Target entity description: Schmidt orthogonalization is a mathematical procedure, also known as the Gram–Schmidt process, that converts a set of linearly independent vectors into an orthonormal set spanning the same subspace.

A. Richardson iteration
Richardson iteration is an early iterative method for solving linear systems and other operator equations, based on repeated relaxation steps to progressively improve an approximate solution.
B. Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
C. Gaussian elimination
Gaussian elimination is a fundamental algorithm in linear algebra used to solve systems of linear equations by systematically transforming matrices into row-echelon form.
D. Jacobi method
The Jacobi method is an iterative numerical algorithm used to solve systems of linear equations by repeatedly updating each variable using values from the previous iteration.
E. Gauss–Seidel method
The Gauss–Seidel method is an iterative numerical technique used to solve systems of linear equations, particularly in large, sparse problems arising in scientific and engineering computations.
F. None of above. chosen

Statements (49)

Predicate	Object
instanceOf	algorithm ⓘ linear algebra concept ⓘ mathematical procedure ⓘ
alternativeName	Gram–Schmidt process ⓘ linked to: Schmidt orthogonalization
appliesTo	Hilbert spaces ⓘ finite-dimensional vector spaces ⓘ
constraint	input vectors must be linearly independent ⓘ
definedOn	Euclidean space ⓘ inner product space ⓘ
ensures	resulting vectors are mutually orthogonal ⓘ resulting vectors have unit norm ⓘ
field	functional analysis ⓘ linear algebra ⓘ
generalizationOf	orthogonalization of two vectors ⓘ
hasVariant	block Gram–Schmidt process ⓘ linked to: Schmidt orthogonalization modified Gram–Schmidt process ⓘ linked to: Schmidt orthogonalization
input	finite sequence of linearly independent vectors ⓘ
mathematicallyDescribedAs	iterative orthogonalization procedure ⓘ
namedAfter	Erhard Schmidt ⓘ Jørgen Pedersen Gram ⓘ
output	orthogonal set of vectors ⓘ orthonormal set of vectors ⓘ
preserves	linear span of the original vectors ⓘ
property	can be numerically unstable in classical form ⓘ order-dependent ⓘ
relatedTo	Cholesky decomposition ⓘ Householder transformation ⓘ QR factorization ⓘ orthogonal projection ⓘ
requires	nonzero starting vectors ⓘ
step	normalize each nonzero vector ⓘ subtract projection onto previously constructed vectors ⓘ
timeComplexity	O(n^2 m) for n vectors in R^m (classical implementation) ⓘ
usedFor	QR decomposition of matrices ⓘ constructing orthonormal bases ⓘ least squares problems ⓘ numerical linear algebra algorithms ⓘ orthogonal polynomial construction ⓘ
usedIn	computational physics ⓘ numerical solutions of differential equations ⓘ signal processing ⓘ statistics ⓘ
usesConcept	inner product ⓘ norm ⓘ orthogonality ⓘ orthonormality ⓘ vector space ⓘ
yields	orthogonal or unitary Q matrix in QR decomposition ⓘ upper triangular R matrix in QR decomposition ⓘ

How these facts were elicited