Aalto Dictionary of ML – Spectral Decomposition

Every normal matrix ${\bf A}\in \mathbb{C}^{d\times d}$ admits a spectral decomposition of the form (Horn and Johnson 2013; Axler 2015) \({\bf A}= \sum_{j=1}^{d} \lambda_{j} {\bf u}^{(j)} \big({\bf u}^{(j)})^{H} \nonumber \\\) with an orthonormal basis ${\bf u}^{(1)},\ldots,{\bf u}^{(d)}$.

{#fig:eigenvectors-length_dict width=”80%”}

Each basis element ${\bf u}^{(j)}$ is an eigenvector of ${\bf A}$ with corresponding eigenvalue $\lambda_{j}$, for $j=1,\ldots,d$.

Axler, Sheldon. 2015. Linear Algebra Done Right. 3rd ed. Cham, Switzerland: Springer Nature.

Horn, R. A., and C. R. Johnson. 2013. Matrix Analysis. 2nd ed. New York, NY, USA: Cambridge Univ. Press.

📚 This explanation is part of the Aalto Dictionary of Machine Learning — an open-access multi-lingual glossary developed at Aalto University to support accessible and precise communication in ML.