Cross-covariance matrix

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In probability theory and statistics, a cross-covariance matrix is a matrix whose element in the i, j position is the covariance between the i-th element of a random vector and j-th element of another 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 cross-covariance matrix generalizes the notion of covariance to multiple dimensions.

The cross-covariance matrix of two random vectors and is typically denoted by or .

Definition[edit]

For random vectors and , each containing random elements whose expected value and variance exist, the cross-covariance matrix of and is defined by[1]:p.336

 

 

 

 

(Eq.1)

where and are vectors containing the expected values of and . The vectors and need not have the same dimension, and either might be a scalar value.

The cross-covariance matrix is the matrix whose entry is the covariance

between the i-th element of and the j-th element of . This gives the following component-wise definition of the cross-covariance matrix.

Example[edit]

For example, if and are random vectors, then is a matrix whose -th entry is .

Properties[edit]

For the cross-covariance matrix, the following basic properties apply:[2]

  1. If and are independent (or somewhat less restrictedly, if every random variable in is uncorrelated with every random variable in ), then

where , and are random vectors, is a random vector, is a vector, is a vector, and are matrices of constants, and is a matrix of zeroes.

Definition for complex random vectors[edit]

If and are complex random vectors, the definition of the cross-covariance matrix is slightly changed. Transposition is replaced by Hermitian transposition:

For complex random vectors, another matrix called the pseudo-cross-covariance matrix is defined as follows:

Uncorrelatedness[edit]

Two random vectors and are called uncorrelated if their cross-covariance matrix matrix is zero.[1]:p.337

Complex random vectors and are called uncorrelated if their covariance matrix and pseudo-covariance matrix is zero, i.e. if .

References[edit]

  1. ^ a b Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBN 978-0-521-86470-1.
  2. ^ Taboga, Marco (2010). "Lectures on probability theory and mathematical statistics".