eigenvalue
C2Technical/Formal (primarily mathematical, scientific, engineering contexts)
Definition
Meaning
In linear algebra, a scalar associated with a linear transformation of a vector space that, when multiplied by a non-zero eigenvector, yields that same eigenvector.
A number (often complex) that reveals fundamental, inherent properties of a matrix or linear operator, indicating factors by which its associated eigenvectors are scaled.
Linguistics
Semantic Notes
Eigenvalues are central to understanding system stability, resonant frequencies, and principal components. The concept is abstract and quantifies intrinsic characteristics independent of the basis or coordinate system.
Dialectal Variation
British vs American Usage
Differences
No significant lexical differences. Both varieties use 'eigenvalue'. The related term 'characteristic value' is an occasional synonym, with no regional preference.
Connotations
Identical technical connotations. Implies advanced mathematical/physics literacy.
Frequency
Equally low-frequency in general discourse, but standard and identical in technical/academic registers in both regions.
Vocabulary
Collocations
Grammar
Valency Patterns
The matrix has an eigenvalue of 5.To find the eigenvalues for the operator.Eigenvalues are associated with eigenvectors.Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “N/A (Technical term not used idiomatically)”
Usage
Context Usage
Business
Rare. Might appear in advanced data science/analytics contexts (e.g., 'PCA relies on eigenvalues to determine principal components').
Academic
Common in mathematics, physics, engineering, computer science, and statistics publications and lectures.
Everyday
Virtually never used in everyday conversation.
Technical
Core term in linear algebra, quantum mechanics, vibration analysis, and machine learning.
Examples
By Part of Speech
verb
British English
- The algorithm is designed to eigenvalue the large sparse matrix efficiently.
- We need to eigenvalue this operator before proceeding.
American English
- The software can eigenvalue the covariance matrix in seconds.
- You cannot directly eigenvalue a non-square matrix.
adverb
British English
- The matrix was analysed eigenvalue-wise.
- The system behaved eigenvalue-dependently.
American English
- The components were sorted eigenvalue-first.
- It converges eigenvalue-rapidly.
adjective
British English
- The eigenvalue distribution was plotted.
- They discussed eigenvalue stability criteria.
American English
- The eigenvalue analysis revealed the system's modes.
- An eigenvalue approach is used in the proof.
Examples
By CEFR Level
- In our maths class, we learned that an eigenvalue is a special number for a matrix.
- The engineer explained that the bridge's natural frequencies are like eigenvalues.
- The stability of the dynamical system is determined by the sign of the largest real part of its eigenvalues.
- Principal Component Analysis works by diagonalising the covariance matrix and ranking the eigenvectors by their corresponding eigenvalues.
- The spectral theorem states that a self-adjoint operator on a Hilbert space has a complete orthonormal basis of eigenvectors with real eigenvalues.
- The Anderson transition in disordered systems is characterised by the localisation of eigenvectors, which is studied via the statistics of their eigenvalues.
Learning
Memory Aids
Mnemonic
Think: 'I generate value' from 'eigen' (German for 'own' or 'characteristic') + 'value'. It's the *characteristic value* that belongs to a specific transformation.
Conceptual Metaphor
THE ESSENCE IS A NUMBER (The eigenvalue is a numerical signature of the system's essential, unchanging behaviour).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Direct calque 'собственное значение' is correct. No trap, but ensure the mathematical concept is understood, not just the translation.
Common Mistakes
- Pronouncing it 'ee-gen-value' or 'egg-en-value' (correct: 'eye-gen-value').
- Confusing eigenvalue with eigenvector.
- Using it as a countable noun incorrectly (e.g., 'an eigenvalues').
Practice
Quiz
What does the dominant eigenvalue of a matrix typically represent in applications like population modelling?
FAQ
Frequently Asked Questions
An eigenvalue is a scalar (number) that scales the eigenvector. An eigenvector is a non-zero vector that, when multiplied by the matrix, only changes by that scalar factor (the eigenvalue), not its direction.
Yes. A zero eigenvalue indicates the matrix is singular (non-invertible). Complex eigenvalues often appear in systems involving rotation or oscillation (e.g., in stability analysis of differential equations).
It comes from German, meaning 'own', 'peculiar to', or 'characteristic'. Thus, eigenvalue means 'characteristic value'.
Extensively. It is fundamental in physics (quantum mechanics, classical mechanics), engineering (vibration analysis, control theory), statistics (Principal Component Analysis), and computer science (graph theory, Google's PageRank algorithm).