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 (that is, of a vector of random variables). Each element of the vector is a scalar random variable, either with a finite number of observed empirical values or with a finite or infinite number of potential values specified by a theoretical joint probability distribution of all the random variables.

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.

Read more about Covariance Matrix: Definition, Conflicting Nomenclatures and Notations, Properties, As A Linear Operator, Which Matrices Are Covariance Matrices?, How To Find A Valid Covariance Matrix, Complex Random Vectors, Estimation, As A Parameter of A Distribution

### Other articles related to "covariance matrix, covariance":

... Several simple measures of spatial dispersion for a point set can be defined using the

**covariance matrix**of the coordinates of the points ... The trace, the determinant, and the largest eigenvalue of the

**covariance matrix**can be used as measures of spatial dispersion ... A measure of spatial dispersion that is not based on the

**covariance matrix**is the average distance between nearest neighbors ...

... The sample mean and the sample

**covariance matrix**are unbiased estimates of the mean and the

**covariance matrix**of the random vector, a row vector whose jth element (j = 1 ... The sample

**covariance matrix**has in the denominator rather than due to a variant of Bessel's correction In short, the sample

**covariance**relies on the difference between each observation and the sample ... The maximum likelihood estimate of the

**covariance**for the Gaussian distribution case has N in the denominator as well ...

... CMA-ES stands for

**Covariance Matrix**Adaptation Evolution Strategy ... this distribution are represented by a

**covariance matrix**... The

**covariance matrix**adaptation (CMA) is a method to update the

**covariance matrix**of this distribution ...

... Using a non-identity

**covariance matrix**for the multivariate normal distribution in evolution strategies is equivalent to a coordinate system transformation of the solution vectors, mainly ... procedure applied to a simple evolution strategy with identity

**covariance matrix**...

**Covariance Matrix**- As A Parameter of A Distribution

... If a vector of n possibly correlated random variables is jointly normally distributed, or more generally elliptically distributed, then its probability density function can be expressed in terms of the covariance matrix. ...

### Famous quotes containing the word matrix:

“As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the *matrix* out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.”

—Margaret Atwood (b. 1939)