Eigenvalue
A scalar by which a matrix stretches or compresses its corresponding eigenvector.
Also: characteristic value
Definition
An eigenvalue is a scalar associated with a square matrix A such that there exists a nonzero vector v (the eigenvector) satisfying the equation Av = (lambda)v, meaning the matrix only scales the vector without changing its direction. Eigenvalues and eigenvectors reveal fundamental properties of linear transformations. They are essential in physics, engineering, machine learning (e.g., principal component analysis), and quantum mechanics.
Example
“Google's PageRank algorithm, which ranks web pages in search results, is computed as the principal eigenvector of the web's link structure matrix, with the largest eigenvalue indicating the most important pages.”
Synonyms
- characteristic value
- spectral value
- matrix eigenvalue
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Related Terms
- Matrix
- Linear Algebra
- Principal Component Analysis
- Quantum Mechanics