<s>
In	O
probability	O
theory	O
and	O
statistics	O
,	O
a	O
cross-covariance	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
whose	O
element	O
in	O
the	O
i	O
,	O
j	O
position	O
is	O
the	O
covariance	O
between	O
the	O
i-th	O
element	O
of	O
a	O
random	B-General_Concept
vector	I-General_Concept
and	O
j-th	O
element	O
of	O
another	O
random	B-General_Concept
vector	I-General_Concept
.	O
</s>
<s>
A	O
random	B-General_Concept
vector	I-General_Concept
is	O
a	O
random	O
variable	O
with	O
multiple	O
dimensions	O
.	O
</s>
<s>
Intuitively	O
,	O
the	O
cross-covariance	B-Algorithm
matrix	I-Algorithm
generalizes	O
the	O
notion	O
of	O
covariance	O
to	O
multiple	O
dimensions	O
.	O
</s>
<s>
The	O
cross-covariance	B-Algorithm
matrix	I-Algorithm
of	O
two	O
random	B-General_Concept
vectors	I-General_Concept
and	O
is	O
typically	O
denoted	O
by	O
or	O
.	O
</s>
<s>
This	O
gives	O
the	O
following	O
component-wise	O
definition	O
of	O
the	O
cross-covariance	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
is	O
a	O
matrix	B-Architecture
whose	O
-th	O
entry	O
is	O
.	O
</s>
<s>
For	O
the	O
cross-covariance	B-Algorithm
matrix	I-Algorithm
,	O
the	O
following	O
basic	O
properties	O
apply	O
:	O
</s>
<s>
where	O
,	O
and	O
are	O
random	O
vectors	O
,	O
is	O
a	O
random	O
vector	O
,	O
is	O
a	O
vector	O
,	O
is	O
a	O
vector	O
,	O
and	O
are	O
matrices	O
of	O
constants	O
,	O
and	O
is	O
a	O
matrix	B-Architecture
of	O
zeroes	O
.	O
</s>
<s>
If	O
and	O
are	O
complex	O
random	B-General_Concept
vectors	I-General_Concept
,	O
the	O
definition	O
of	O
the	O
cross-covariance	B-Algorithm
matrix	I-Algorithm
is	O
slightly	O
changed	O
.	O
</s>
<s>
Transposition	O
is	O
replaced	O
by	O
Hermitian	B-Algorithm
transposition	I-Algorithm
:	O
</s>
<s>
For	O
complex	O
random	B-General_Concept
vectors	I-General_Concept
,	O
another	O
matrix	B-Architecture
called	O
the	O
pseudo-cross-covariance	O
matrix	B-Architecture
is	O
defined	O
as	O
follows	O
:	O
</s>
<s>
Two	O
random	B-General_Concept
vectors	I-General_Concept
and	O
are	O
called	O
uncorrelated	O
if	O
their	O
cross-covariance	B-Algorithm
matrix	I-Algorithm
matrix	B-Architecture
is	O
a	O
zero	O
matrix	B-Architecture
.	O
</s>
<s>
Complex	O
random	B-General_Concept
vectors	I-General_Concept
and	O
are	O
called	O
uncorrelated	O
if	O
their	O
covariance	O
matrix	B-Architecture
and	O
pseudo-covariance	O
matrix	B-Architecture
is	O
zero	O
,	O
i.e.	O
</s>
