<s>
In	O
linear	B-Language
algebra	I-Language
and	O
statistics	O
,	O
the	O
pseudo-determinant	B-Algorithm
is	O
the	O
product	O
of	O
all	O
non-zero	O
eigenvalues	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
pseudo-determinant	B-Algorithm
of	O
a	O
square	O
n-by-n	O
matrix	O
A	O
may	O
be	O
defined	O
as	O
:	O
</s>
<s>
where	O
|A|	O
denotes	O
the	O
usual	O
determinant	O
,	O
I	O
denotes	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
and	O
rank(A )	O
denotes	O
the	O
rank	O
of	O
A	O
.	O
</s>
<s>
If	O
all	O
singular	O
values	O
are	O
zero	O
,	O
then	O
the	O
pseudo-determinant	B-Algorithm
is1	O
.	O
</s>
<s>
Supposing	O
,	O
so	O
that	O
k	O
is	O
the	O
number	O
of	O
non-zero	O
singular	O
values	O
,	O
we	O
may	O
write	O
where	O
is	O
some	O
n-by-k	O
matrix	O
and	O
the	O
dagger	O
is	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
.	O
</s>
<s>
If	O
a	O
statistical	O
procedure	O
ordinarily	O
compares	O
distributions	O
in	O
terms	O
of	O
the	O
determinants	O
of	O
variance-covariance	O
matrices	O
then	O
,	O
in	O
the	O
case	O
of	O
singular	O
matrices	O
,	O
this	O
comparison	O
can	O
be	O
undertaken	O
by	O
using	O
a	O
combination	O
of	O
the	O
ranks	O
of	O
the	O
matrices	O
and	O
their	O
pseudo-determinants	B-Algorithm
,	O
with	O
the	O
matrix	O
of	O
higher	O
rank	O
being	O
counted	O
as	O
"	O
largest	O
"	O
and	O
the	O
pseudo-determinants	B-Algorithm
only	O
being	O
used	O
if	O
the	O
ranks	O
are	O
equal	O
.	O
</s>
<s>
Thus	O
pseudo-determinants	B-Algorithm
are	O
sometime	O
presented	O
in	O
the	O
outputs	O
of	O
statistical	O
programs	O
in	O
cases	O
where	O
covariance	O
matrices	O
are	O
singular	O
.	O
</s>
