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
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