Dictionary of Applied Machine Learning
Typeset PDF version — the authoritative form of this entry
The covariance matrix of a random vector is the expectation of the outer product of the centered random vector with itself, i.e., of the random vector minus its mean. Its entry in row $\featureidx$ and column $\featureidx'$ is the covariance between the corresponding entries of the random vector. In particular, the diagonal entries are the variances of the individual entries. The covariance matrix thus collects the second-order statistics of a random vector.
The covariance matrix of a random vector $\featurevec \in \mathbb{R}^{\featuredim}$ is defined as the expectation (if it exists): \[\covmtx{\featurevec} \defeq \expect \bigg \{ \big( \featurevec - \expect \big\{ \featurevec \big\} \big) \big(\featurevec - \expect \big\{ \featurevec \big\} \big)^{\top} \bigg\}.\]
See also: covariance, matrix, random variable, random vector.
@misc{dictml_covmtx,
author = {Jung, Alexander},
title = {covariance matrix},
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/covmtx.html}
}