gram-schmidt orthogonalization: meaning, definition, pronunciation and examples
C2Technical/Academic
Quick answer
What does “gram-schmidt orthogonalization” mean?
A mathematical procedure that takes a finite, linearly independent set of vectors and produces an orthogonal set that spans the same subspace.
Audio
Pronunciation
Definition
Meaning and Definition
A mathematical procedure that takes a finite, linearly independent set of vectors and produces an orthogonal set that spans the same subspace.
A fundamental algorithm in linear algebra used to construct an orthonormal basis from any set of vectors, commonly applied in numerical analysis, signal processing, and data science for tasks like QR decomposition.
Dialectal Variation
British vs American Usage
Differences
Primarily spelling: British English uses 'orthogonalisation', American English uses 'orthogonalization'. The en dash in 'Gram–Schmidt' is often rendered as a hyphen or space in informal writing.
Connotations
None beyond standard technical register.
Frequency
Equally frequent in relevant academic/technical fields in both regions.
Grammar
How to Use “gram-schmidt orthogonalization” in a Sentence
Gram–Schmidt orthogonalization is performed on [a set of vectors].One can orthogonalize [the basis] via Gram–Schmidt.Applying Gram–Schmidt orthogonalization yields [an orthonormal basis].Vocabulary
Collocations
Examples
Examples of “gram-schmidt orthogonalization” in a Sentence
verb
British English
- We need to orthogonalise the basis using the Gram–Schmidt method.
- The algorithm orthogonalises the input vectors.
American English
- We need to orthogonalize the basis using the Gram–Schmidt method.
- The algorithm orthogonalizes the input vectors.
adjective
British English
- The Gram–Schmidt orthogonalised basis is more stable.
- Follow the Gram–Schmidt orthogonalisation steps.
American English
- The Gram–Schmidt orthogonalized basis is more stable.
- Follow the Gram–Schmidt orthogonalization steps.
Usage
Meaning in Context
Business
Virtually never used.
Academic
Core term in mathematics, physics, engineering, and computer science courses on linear algebra.
Everyday
Not used.
Technical
Essential in numerical linear algebra, quantum mechanics, machine learning (e.g., in principal component analysis pre-processing).
Vocabulary
Synonyms of “gram-schmidt orthogonalization”
Strong
Neutral
Weak
Vocabulary
Antonyms of “gram-schmidt orthogonalization”
Watch out
Common Mistakes When Using “gram-schmidt orthogonalization”
- Misspelling as 'Graham-Schmidt' or 'Gram-Schmidt orthogonalisation' in American English.
- Using it as a verb incorrectly: 'We Gram–Schmidt the vectors.' (Preferred: 'We apply Gram–Schmidt to the vectors.')
FAQ
Frequently Asked Questions
Yes, these terms are used interchangeably to refer to the same algorithm.
Gram–Schmidt produces an orthogonal set. If the vectors are also normalized to unit length, the result is an orthonormal set. The process is often described as 'orthogonalization', with normalization as a final, separate step.
Jørgen Pedersen Gram (Danish) and Erhard Schmidt (German) were mathematicians who independently developed the procedure in the context of inner product spaces.
It is fundamental in computer graphics, machine learning (e.g., orthogonal weight initialization), signal processing, and solving large systems of equations via QR decomposition.
A mathematical procedure that takes a finite, linearly independent set of vectors and produces an orthogonal set that spans the same subspace.
Gram-schmidt orthogonalization is usually technical/academic in register.
Gram-schmidt orthogonalization: in British English it is pronounced /ɡram ʃmɪt ɔːˌθɒɡənəlaɪˈzeɪʃən/, and in American English it is pronounced /ɡræm ʃmɪt ɔːrˌθɑːɡənələˈzeɪʃən/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think: 'Graham and Schmidt' are two mathematicians who 'straighten' (orthogonalize) a messy set of vectors into neat, perpendicular ones.
Conceptual Metaphor
A RECIPE for purifying a mixture of ingredients (vectors) into distinct, non-overlapping components.
Practice
Quiz
What is the primary purpose of Gram–Schmidt orthogonalization?