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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 (*i*, *j*) entry is the covariance

where

is the expected value of the *i*th 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:

- is positive semi-definite
- If
*p*=*q*, then - 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 *M*^{1/2}. 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.

## Further reading[]

- Covariance Matrix at MathWorld
- van Kampen, N. G.
*Stochastic processes in physics and chemistry*. New York: North-Holland, 1981.

## See also[]

- Estimation of covariance matrices
- Multivariate statistics
- Sample covariance matrix
- Gramian matrix

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