eigenvalue

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An eigenvalue of a square matrix $\mA$ is a number $\eigvalgen$ for which some nonzero vector $\vx$ satisfies $\mA\vx = \eigvalgen\vx$. Such a vector is referred to as an eigenvector of $\mA$. Multiplying an eigenvector by the matrix scales it by the eigenvalue without changing its direction. The eigenvalues of a diagonalizable matrix constitute the diagonal factor of its eigenvalue decomposition (EVD).

Definition

A number $\eigvalgen \in \mathbb{R}$ is called an eigenvalue of a square matrix $\mathbf{A} \in \mathbb{R}^{\featuredim \times \featuredim}$ if there exists a nonzero vector $\vx \in \mathbb{R}^{\featuredim} \setminus \{ \mathbf{0} \}$ such that $\mathbf{A} \vx = \eigvalgen \vx$ (see Fig.\ 1).

Figure 1 of the entry eigenvalue — Figure 1: Eigenvector corresponding to the eigenvalue $\eigvalgen$.

Cite this entry

@misc{dictml_eigenvalue,
  author = {Jung, Alexander},
  title = {eigenvalue},
  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/eigenvalue.html}
}