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 .

DefinitionEdit

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.

 

ExampleEdit

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

PropertiesEdit

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

  1.  
  2.  
  3.  
  4.  
  5. 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 vectorsEdit

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:

 

UncorrelatednessEdit

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

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

ReferencesEdit

  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".