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 formula_1 and formula_2 is typically denoted by formula_3 or formula_4.
Definition.
For random vectors formula_1 and formula_2, each containing random elements whose expected value and variance exist, the cross-covariance matrix of formula_1 and formula_2 is defined by
where formula_9 and formula_10 are vectors containing the expected values of formula_1 and formula_2. The vectors formula_1 and formula_2 need not have the same dimension, and either might be a scalar value.
The cross-covariance matrix is the matrix whose formula_15 entry is the covariance
between the "i"-th element of formula_1 and the "j"-th element of formula_2. This gives the following component-wise definition of the cross-covariance matrix.
Example.
For example, if formula_20 and formula_21 are random vectors, then
formula_22 is a formula_23 matrix whose formula_15-th entry is formula_25.
Properties.
For the cross-covariance matrix, the following basic properties apply:
where formula_1, formula_36 and formula_37 are random formula_38 vectors, formula_2 is a random formula_40 vector, formula_41 is a formula_40 vector, formula_43 is a formula_38 vector, formula_45 and formula_46 are formula_47 matrices of constants, and formula_48 is a formula_49 matrix of zeroes.
Definition for complex random vectors.
If formula_50 and formula_51 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 formula_1 and formula_2 are called uncorrelated if their cross-covariance matrix formula_3 matrix is a zero matrix.
Complex random vectors formula_50 and formula_51 are called uncorrelated if their covariance matrix and pseudo-covariance matrix is zero, i.e. if formula_59.