Covariance Matrix - As A Linear Operator

As A Linear Operator

Applied to one vector, the covariance matrix maps a linear combination, c, of the random variables, X, onto a vector of covariances with those variables: . Treated as a bilinear form, it yields the covariance between the two linear combinations: . The variance of a linear combination is then, its covariance with itself.

Similarly, the (pseudo-)inverse covariance matrix provides an inner product, which induces the Mahalanobis distance, a measure of the "unlikelihood" of c.