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In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. It is the natural generalization to higher dimensions of the concept of the variance of a scalar-valued random variable.

## Definition

If entries in the column vector

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

where

is the expected value of the ith entry in the vector X. In other words, we have

### As a generalization of the variance

The definition above is equivalent to the matrix equality

This form can be seen as a generalization of the scalar-valued variance to higher dimensions. Recall that for a scalar-valued random variable X

where

The matrix is also often called the variance-covariance matrix since the diagonal terms are in fact variances.

## Conflicting nomenclatures and notations

Nomenclatures differ. Some statisticians, following the probabilist William Feller, call this matrix the variance of the random vector , because it is the natural generalization to higher dimensions of the 1-dimensional variance. Others call it the covariance matrix, because it is the matrix of covariances between the scalar components of the vector . Thus

However, the notation for the "cross-covariance" between two vectors is standard:

The notation is found in William Feller's two-volume book An Introduction to Probability Theory and Its Applications, but both forms are quite standard and there is no ambiguity between them.

## Properties

For and the following basic properties apply:

1. is positive semi-definite

2. If p = q, then

3. If and are independent, then

where and are a random vectors, is a random vector, is vector, and are matrices.

This covariance matrix (though very simple) is a very useful tool in many very different areas. From it a transformation matrix can be derived that allows one to completely decorrelate the data or, from a different point of view, to find an optimal basis for representing the data in a compact way (see Rayleigh quotient for a formal proof and additional properties of covariance matrices). This is called principal components analysis (PCA) in statistics and Karhunen-Loève transform (KL-transform) in image processing.

## Which matrices are covariance matrices

From the identity

and the fact that the variance of any real-valued random variable is nonnegative, it follows immediately that only a nonnegative-definite matrix can be a covariance matrix. The converse question is whether every nonnegative-definite symmetric matrix is a covariance matrix. The answer is "yes". To see this, suppose M is a p×p nonnegative-definite symmetric matrix. From the finite-dimensional case of the spectral theorem, it follows that M has a nonnegative symmetric square root, which let us call M1/2. Let be any p×1 column vector-valued random variable whose covariance matrix is the p×p identity matrix. Then

## Complex random vectors

The variance of a complex scalar-valued random variable with expected value μ is conventionally defined using complex conjugation:

where the complex conjugate of a complex number is denoted .

If is a column-vector of complex-valued random variables, then we take the conjugate transpose by both transposing and conjugating, getting a square matrix:

where denotes the conjugate transpose, which is applicable to the scalar case since the transpose of a scalar is still a scalar.

LaTeX provides useful features for dealing with covariance matrices. These are available through the extendedmath package.

## Estimation

The derivation of the maximum-likelihood estimator of the covariance matrix of a multivariate normal distribution is perhaps surprisingly subtle. It involves the spectral theorem and the reason why it can be better to view a scalar as the trace of a 1 × 1 matrix than as a mere scalar. See estimation of covariance matrices.