Co-difference matrix

concept

provided

is an n-dimensional random variable, said matrix

n-dimensional random variable

covariance matrix (covariance matrix), also referred to as

, where

to

component

and

covariance (provided they are present).

For example, a two-dimensional random variable

of the covariance matrix

where

Since the

, so the covariance matrix is ​​nonnegative definite symmetric matrix.

properties

covariance matrix with the following properties:

(1)

(2).

, where A is a matrix, b is a vector.

(3)

.

Applications

covariance matrix may be used to represent probability density multidimensional random variables, thus the study of multidimensional random variables can be achieved by the covariance matrix. A two-dimensional random variable

as an example, since the

matrix incorporated

,

and

covariance matrix

thus available

Since

Thus

of the probability density

this equation can be generalized to an n-dimensional normal distribution.