Dictionary of Applied Machine Learning
Typeset PDF version — the authoritative form of this entry
An eigenvalue decomposition (EVD) is a factorization of a square matrix of the form $\mA = \mV \bm{\Lambda} \mV^{-1}$. The columns of the matrix $\mV$ are eigenvectors of $\mA$. The diagonal matrix $\bm{\Lambda}$ contains the eigenvalues corresponding to these eigenvectors. A matrix that admits an EVD is referred to as diagonalizable.
An EVD for a square matrix $\mA \in \mathbb{R}^{\dimlocalmodel \times \dimlocalmodel}$ is a factorization of the form \[\mA = \mathbf{V} {\bm \Lambda} \mathbf{V}^{-1}.\] The columns of the matrix $\mV = \big( \vv^{(1)}, \,\ldots, \,\vv^{(\dimlocalmodel)} \big)$ are the eigenvectors of the matrix $\mA$. The diagonal matrix ${\bm \Lambda} = {\rm diag} \big\{ \eigval{1}, \,\ldots, \,\eigval{\dimlocalmodel} \big\}$ contains the eigenvalues $\eigval{\featureidx}$ corresponding to the eigenvectors $\vv^{(\featureidx)}$. Matrices that allow for an EVD are referred to as diagonalizable.
See also: matrix, eigenvector, eigenvalue, diagonalizable.
@misc{dictml_evd,
author = {Jung, Alexander},
title = {eigenvalue decomposition},
howpublished = {Dictionary of Applied Machine Learning (course edition)},
year = {2026},
doi = {10.5281/zenodo.21569296},
note = {ISBN 978-952-64-3013-3, CC BY 4.0, retrieved 2026-08-06},
url = {https://dictionaryofml.org/terms/evd.html}
}