Covariance matrices

In probability theory and statistics, a covariance matrix (also known as dispersion matrix or variance–covariance matrix) is a matrix whose element in the i, j position is the covariance between the i ^th and j ^th elements of a random vector. A random vector is a random variable with multiple dimensions. Each element of the vector is a scalar random variable. Each element has either a finite number of observed empirical values or a finite or infinite number of potential values. The potential values are specified by a theoretical joint probability distribution.

Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the x and y directions contain all of the necessary information; a 2×2 matrix would be necessary to fully characterize the two-dimensional variation.

Because the covariance of the i^th random variable with itself is simply that random variable's variance, each element on the principal diagonal of the covariance matrix is the variance of one of the random variables. Because the covariance of the i^th random variable with the j^th one is the same thing as the covariance of the j^th random variable with the i^th one, every covariance matrix is symmetric. In addition, every covariance matrix is positive semi-definite.

Throughout this article, boldfaced unsubscripted X and Y are used to refer to random vectors, and unboldfaced subscripted X_i and Y_i are used to refer to random scalars.

If the entries in the column vector

are random variables, each with finite variance, then the covariance matrix Σ is the matrix whose (i, j) entry is the covariance

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