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[[Image:Gaussian-2d.png|Gaussian-2d|thumb]] {{Short description|Matrix of covariances between elements of a random vector}}
{{Linear algebra}}
{{Probability theory}}

A '''covariance matrix''' is a square matrix that contains the [[covariance]]s between elements of a [[random vector]]. It is a key concept in [[probability theory]] and [[statistics]], particularly in the fields of [[multivariate statistics]] and [[linear algebra]].

== Definition ==
Given a random vector \(\mathbf{X} = \begin{bmatrix} X_1 & X_2 & \cdots & X_n \end{bmatrix}^T\) of \(n\) random variables, the covariance matrix \(\Sigma\) is defined as:
\[
\Sigma = \text{Cov}(\mathbf{X}) = \mathbb{E} \left[ (\mathbf{X} - \mathbb{E}[\mathbf{X}])(\mathbf{X} - \mathbb{E}[\mathbf{X}])^T \right]
\]
where \(\mathbb{E}[\mathbf{X}]\) is the [[expected value]] (mean) vector of \(\mathbf{X}\).

== Properties ==
* **Symmetry**: The covariance matrix is symmetric, i.e., \(\Sigma = \Sigma^T\).
* **Positive Semi-Definiteness**: The covariance matrix is positive semi-definite, meaning that for any non-zero vector \(\mathbf{a}\), \(\mathbf{a}^T \Sigma \mathbf{a} \geq 0\).
* **Diagonal Elements**: The diagonal elements of the covariance matrix are the variances of the individual random variables, i.e., \(\Sigma_{ii} = \text{Var}(X_i)\).

== Applications ==
Covariance matrices are used in various applications, including:
* **[[Principal component analysis]] (PCA)**: PCA uses the covariance matrix to identify the principal components of the data.
* **[[Portfolio theory]]**: In finance, the covariance matrix is used to model the returns of different assets and to optimize the portfolio.
* **[[Kalman filter]]**: The covariance matrix is used in the Kalman filter algorithm to estimate the state of a dynamic system.

== Example ==
Consider a random vector \(\mathbf{X} = \begin{bmatrix} X_1 & X_2 \end{bmatrix}^T\) with the following properties:
\[
\mathbb{E}[\mathbf{X}] = \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \quad \text{Cov}(X_1, X_2) = \sigma_{12}
\]
The covariance matrix \(\Sigma\) is:
\[
\Sigma = \begin{bmatrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{bmatrix}
\]
where \(\sigma_1^2\) and \(\sigma_2^2\) are the variances of \(X_1\) and \(X_2\), respectively.

== See also ==
* [[Variance]]
* [[Correlation matrix]]
* [[Multivariate normal distribution]]
* [[Linear regression]]
* [[Eigenvalues and eigenvectors]]

== Related pages ==
* [[Probability theory]]
* [[Statistics]]
* [[Linear algebra]]
* [[Principal component analysis]]
* [[Portfolio theory]]
* [[Kalman filter]]

[[Category:Linear algebra]]
[[Category:Probability theory]]
[[Category:Statistics]]
[[Category:Matrices]]

{{Linear-algebra-stub}}

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Prab: CSV import