Information Theory and Principal Component Analysis - Biomedical Signal Analysis: Contemporary Methods and Applications - page 80

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

2

1 2

1

0

0

1

1

2

2

Figure 3.2
Density of a two-dimensional normal distribution i.e. a Gaussian with zero mean
and unit variance.

The density of a two-dimensional Gaussian is shown in figure 3.2.
Note that a Gaussian random vector is independent if and only if it
is decorrelated. Only mean and variance are needed to describe Gaus-
sians, so it is not surprising that detection of second-order information
(decorrelation) already leads to independence. Furthermore, note that
the conditional density of a Gaussian is again Gaussian.

Lemma 3.6: Let X be a Gaussian n -dimensional random vector and

let A Gl( n ). Then AX is Gaussian. If X is independent, then AX is

independent if and only if A

∈

∈ O ( n ).

Proof The first- and second-order moments of X do not change by
being multiplied by an orthogonal matrix, so if A

O ( n ), then AX
is independent. If, however, AX is independent, then I =Cov( X )=
A Cov( X ) A = AA ,so A

∈

∈ O ( n ).