A $k\times k$ matrix of covariances between all pairs of $k$ random variables. It is also called variance-covariance matrix or simply the covariance matrix.

Given a random vector $\mathbf x$ with $k$ elements, its covariance matrix (also called variance-covariance matrix or simply variance matrix) is a $k \times k$ matrix $\mathbf C$ of covariances between all pairs of elements: $C_{ij} = \operatorname{Cov}(x_i, x_j)$. Diagonal elements of the covariance matrix are variances of each $x_i$. Covariance matrix is always symmetric and positive semi-definite.

The sample covariance matrix, for a random vector $\mathbf{x}$ is also a $k \times k$ matrix:

$$Q = \frac{1}{n-1} \sum_{i=1}^{n}(x_{i} - \bar{x})(x_{i} - \bar{x})^T$$

The sample covariance matrix is an unbiased estimate of the covariance matrix. Both covariance and sample-covariance matrices are positive semi-definite.

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