# Random Vectors

Vectors :: Covariance :: Functions :: Application :: Markov Model

## Covariance

Suppose
and
(that is,
and
are
random vectors of dimension
and
, respectively).

Note:
is frequently used as a symbol to denote covariance.
It should not be confused with a summation sign, and is usually clear
from context.

Property:

If
is
and
is
and
and
then

is called the ``covariance of .'' It is a symmetric matrix, non-negative definite (or positive semidefinite), and thus has all non-negative eigenvalues.

If
are mutually uncorrelated then

where

Suppose we partition
of
dimensions as

of and elements, respectively. let

where and . Similarly,

where

or, in general,