Correlatiematrix

Correlatiematrix, or correlation matrix, is a symmetric p×p matrix whose (i,j) entry is the correlation coefficient between random variables X_i and X_j. For a vector X = (X_1, ..., X_p), the population correlation matrix R has R_ii = 1 and R_ij = Corr(X_i, X_j) = Cov(X_i, X_j) / (σ_i σ_j), where σ_i is the standard deviation of X_i. Because it is the Gram matrix of standardized variables, R is positive semidefinite.

In practice, the correlation matrix is estimated from data. Given n observations of p variables, form the n×p data matrix X, usually standardizing variables to zero mean and unit variance, and compute the sample correlation matrix R̂ with entries r̂_ij = cov̂(X_i, X_j) / (s_i s_j). The diagonal entries are 1, the off-diagonal entries lie in [-1, 1], and R̂ is symmetric.

Properties and interpretation: The correlation matrix summarizes linear relationships among variables. It is widely used in multivariate analysis, including principal component analysis, factor analysis, and multivariate modeling. In the special case of multivariate normal data, the correlation matrix, together with means, fully describes the dependence structure. Note that correlations measure linear association and can be influenced by outliers; they do not imply causation and may miss nonlinear relationships.

Relation to the covariance matrix: Cov(X) = D R D, where D is the diagonal matrix of standard deviations. Variants and extensions include partial correlation matrices, Spearman or rank-based correlation matrices, and robust or shrinkage estimators used in high-dimensional settings.