eigenvector
C1/C2 (Specialized/Technical)Formal, Academic, Technical (primarily used in mathematics, physics, engineering, computer science, and data science contexts)
Definition
Meaning
A nonzero vector that, when a linear transformation is applied to it, changes only by a scalar factor; a vector whose direction remains unchanged when multiplied by a given matrix.
In mathematics, physics, and engineering, an eigenvector represents a direction that is invariant under a given transformation; in data science and machine learning, eigenvectors of covariance matrices (e.g., in PCA) indicate directions of maximum variance in the data.
Linguistics
Semantic Notes
The term is inherently mathematical. Its meaning is precise and does not vary by context, though its applications span multiple disciplines. The concept is central to linear algebra and spectral theory.
Dialectal Variation
British vs American Usage
Differences
No lexical or spelling differences. Pronunciations differ slightly (see IPA). Conceptual usage is identical across dialects.
Connotations
Highly technical term with no colloquial connotations. Associated with advanced mathematics, quantum mechanics, and data science.
Frequency
Used with identical frequency in technical/academic writing in both regions. Virtually absent from general everyday speech.
Vocabulary
Collocations
Grammar
Valency Patterns
eigenvector of [matrix]eigenvector corresponding to [eigenvalue]eigenvector for [transformation]eigenvector associated witheigenvector decomposition ofVocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “None. The term is strictly technical.”
Usage
Context Usage
Business
Rare, except in contexts like data analytics, finance (portfolio optimization, risk modeling), or tech startups specializing in AI/ML.
Academic
Core term in linear algebra, quantum mechanics, vibration analysis, statistics (PCA), and computer graphics.
Everyday
Virtually never used in everyday conversation.
Technical
Fundamental in mathematics, physics, engineering (structural, electrical), computer science (algorithms, PageRank), and data science.
Examples
By Part of Speech
verb
British English
- The algorithm will eigen-decompose the matrix.
- We need to diagonalise the operator to find its eigenvectors.
American English
- The software eigen-solves the system.
- We diagonalize the matrix to compute its eigenvectors.
adverb
British English
- The matrix acts eigenvector-wise only for those specific directions.
- The data was transformed eigenvector-ally.
American English
- The system responds eigenvector-wise under that transformation.
- The components were sorted eigenvector-ally.
adjective
British English
- The eigenvector solution is not unique.
- They performed an eigenvector analysis on the dataset.
American English
- The eigenvector computation is stable.
- This is an eigenvector-based method for dimensionality reduction.
Examples
By CEFR Level
- Not applicable for A2 level.
- In our maths class, we learned that an eigenvector doesn't change direction when you multiply it by a matrix.
- The principal eigenvector of the connectivity matrix helps identify the most influential node in the network.
- Principal Component Analysis relies on calculating the eigenvectors of the covariance matrix to identify the axes of maximum variance in high-dimensional data.
Learning
Memory Aids
Mnemonic
Think of 'Eigen' as 'own' or 'characteristic' in German. An eigenvector is a vector that is 'characteristic' of a matrix — it stays on its own line when the matrix acts on it.
Conceptual Metaphor
The direction of an eigenvector is like a 'pure mode' of a system — for example, the natural sway direction of a bridge or the primary axis of variance in a dataset.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Do not confuse with 'собственный вектор' (the direct translation is correct). Ensure understanding of the underlying mathematical concept, not just the word.
Common Mistakes
- Misspelling as 'eigen vector' (should be one word or hyphenated: eigenvector or eigen-vector).
- Confusing eigenvector with eigenvalue (the vector vs. the scalar factor).
- Using it in non-mathematical contexts where it is inappropriate.
Practice
Quiz
What is the defining property of an eigenvector v for a square matrix A?
FAQ
Frequently Asked Questions
An eigenvector is the vector whose direction is unchanged by a linear transformation. The eigenvalue is the scalar factor by which the eigenvector is scaled when the transformation is applied.
Yes. A matrix typically has a set of eigenvectors, each associated with a specific eigenvalue. There can be multiple linearly independent eigenvectors for a single eigenvalue (forming an eigenspace).
Not always. Eigenvectors of a general matrix are not guaranteed to be orthogonal. However, eigenvectors of a symmetric (or Hermitian) matrix corresponding to distinct eigenvalues are orthogonal.
Eigenvectors are fundamental to techniques like Principal Component Analysis (PCA) and Singular Value Decomposition (SVD). They identify the underlying, uncorrelated directions of maximum variance or structure in complex datasets, enabling dimensionality reduction and feature extraction.