Cross-covariance matrix


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

For random vectors and, each containing random elements whose expected value and variance exist, the cross-covariance matrix of and is defined by
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

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

Properties

For the cross-covariance matrix, the following basic properties apply:
  5. If and are independent, 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

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

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