For a Sequence of Random variables $(X_1,...,X_n)$ (e.g. a random vector), the covariance matrix is the matrix of Covariances, i.e. the $(i,j)$th element of the matrix is the covariance between variable $X_i$ and $X_j$.

Empirical covariance matrix

The matrix of empirical covariances (where the Expectation in the definition of the covariance is substituted by the Sample mean)

some properties of positive covariance matrices: http://www.numdam.org/article/SAD_1988__13_1_8_0.pdf , using Perron-Frobenius theorem