Schmidt orthogonalization
E384564
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| 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 |
Schmidt orthogonalization
ⓘ
surface form:
Gram–Schmidt process
|
| 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 |
Schmidt orthogonalization
self-linksurface differs
ⓘ
surface form:
block Gram–Schmidt process
Schmidt orthogonalization self-linksurface differs ⓘ
surface form:
modified Gram–Schmidt process
|
| input | finite sequence of linearly independent vectors ⓘ |
| mathematicallyDescribedAs | iterative orthogonalization procedure ⓘ |
| namedAfter |
Erhard Schmidt
NERFINISHED
ⓘ
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Schmidt orthogonalization Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Gram–Schmidt process
this entity surface form:
modified Gram–Schmidt process
this entity surface form:
block Gram–Schmidt process