covariantiematrix

The covariantiematrix, more commonly called the covariance matrix, is a matrix that summarizes how a set of random variables vary together. It captures the pairwise linear relationships between the components of a random vector and is fundamental in multivariate statistics and data analysis.

Formally, for a random vector X = (X1, ..., Xn) with mean vector mu, the covariance matrix Sigma is defined as Sigma = E[(X - mu)(X - mu)^T]. Equivalently, Sigma can be written as E[XX^T] - mu mu^T. The matrix is symmetric (Sigma = Sigma^T) and positive semidefinite. Its diagonal entries are the variances Var(Xi), and the off-diagonal entries Cov(Xi, Xj) measure the covariances between different components. If the variables are centered (mean zero), Sigma = E[XX^T].

In practice, the population covariance is often unknown and estimated from data by the sample covariance matrix. Given observations x1, ..., xn of the vector X, the sample covariance is S = (1/(n-1)) sum_k (xk - x_bar)(xk - x_bar)^T, where x_bar is the sample mean. S is symmetric and positive semidefinite; it is invertible if the observed variables are linearly independent.

Applications of the covariantiematrix include principal component analysis, Mahalanobis distance, and multivariate normal modeling. It provides a basis for understanding dependence structure, performing dimensionality reduction, and designing multivariate statistical procedures. A standardized version of Sigma yields the correlation matrix, where diagonal entries are 1 and off-diagonals are correlations.