<s>
In	O
mathematics	O
,	O
an	O
EP	B-Algorithm
matrix	I-Algorithm
(	O
or	O
range-Hermitian	B-Algorithm
matrix	I-Algorithm
or	O
RPN	B-Algorithm
matrix	I-Algorithm
)	O
is	O
a	O
square	O
matrix	O
A	O
whose	O
range	O
is	O
equal	O
to	O
the	O
range	O
of	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
A*	O
.	O
</s>
<s>
Another	O
equivalent	O
characterization	O
of	O
EP	B-Algorithm
matrices	I-Algorithm
is	O
that	O
the	O
range	O
of	O
A	O
is	O
orthogonal	O
to	O
the	O
nullspace	O
of	O
A	O
.	O
</s>
<s>
Thus	O
,	O
EP	B-Algorithm
matrices	I-Algorithm
are	O
also	O
known	O
as	O
RPN	O
(	O
Range	O
Perpendicular	O
to	O
Nullspace	O
)	O
matrices	O
.	O
</s>
<s>
EP	B-Algorithm
matrices	I-Algorithm
were	O
introduced	O
in	O
1950	O
by	O
Hans	O
Schwerdtfeger	O
,	O
and	O
since	O
then	O
,	O
many	O
equivalent	O
characterizations	O
of	O
EP	B-Algorithm
matrices	I-Algorithm
have	O
been	O
investigated	O
through	O
the	O
literature	O
.	O
</s>
<s>
The	O
meaning	O
of	O
the	O
EP	O
abbreviation	O
stands	O
originally	O
for	O
Equal	O
Principal	O
,	O
but	O
it	O
is	O
widely	O
believed	O
that	O
it	O
stands	O
for	O
Equal	O
Projectors	B-Algorithm
instead	O
,	O
since	O
an	O
equivalent	O
characterization	O
of	O
EP	B-Algorithm
matrices	I-Algorithm
is	O
based	O
in	O
terms	O
of	O
equality	O
of	O
the	O
projectors	B-Algorithm
AA+	O
and	O
A+A	O
.	O
</s>
<s>
When	O
A	O
is	O
an	O
EP	B-Algorithm
matrix	I-Algorithm
,	O
the	O
range	O
of	O
A	O
is	O
precisely	O
perpendicular	O
to	O
the	O
null-space	O
of	O
A	O
.	O
</s>
<s>
An	O
equivalent	O
characterization	O
of	O
an	O
EP	B-Algorithm
matrix	I-Algorithm
A	O
is	O
that	O
A	O
commutes	O
with	O
its	O
Moore-Penrose	O
inverse	O
,	O
that	O
is	O
,	O
the	O
projectors	B-Algorithm
AA+	O
and	O
A+A	O
are	O
equal	O
.	O
</s>
<s>
This	O
is	O
similar	O
to	O
the	O
characterization	O
of	O
normal	O
matrices	O
where	O
A	O
commutes	O
with	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
.	O
</s>
<s>
As	O
a	O
corollary	O
,	O
nonsingular	O
matrices	O
are	O
always	O
EP	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
The	O
sum	O
of	O
EP	B-Algorithm
matrices	I-Algorithm
Ai	O
is	O
an	O
EP	B-Algorithm
matrix	I-Algorithm
if	O
the	O
null-space	O
of	O
the	O
sum	O
is	O
contained	O
in	O
the	O
null-space	O
of	O
each	O
matrix	O
Ai	O
.	O
</s>
<s>
To	O
be	O
an	O
EP	B-Algorithm
matrix	I-Algorithm
is	O
a	O
necessary	O
condition	O
for	O
normality	O
:	O
A	O
is	O
normal	O
if	O
and	O
only	O
if	O
A	O
is	O
EP	B-Algorithm
matrix	I-Algorithm
and	O
AA*A2	O
=	O
A2A*A	O
.	O
</s>
<s>
When	O
A	O
is	O
an	O
EP	B-Algorithm
matrix	I-Algorithm
,	O
the	O
Moore-Penrose	O
inverse	O
of	O
A	O
is	O
equal	O
to	O
the	O
group	B-Algorithm
inverse	I-Algorithm
of	O
A	O
.	O
</s>
<s>
A	O
is	O
an	O
EP	B-Algorithm
matrix	I-Algorithm
if	O
and	O
only	O
if	O
the	O
Moore-Penrose	O
inverse	O
of	O
A	O
is	O
an	O
EP	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
spectral	O
theorem	O
states	O
that	O
a	O
matrix	O
is	O
normal	O
if	O
and	O
only	O
if	O
it	O
is	O
unitarily	B-Algorithm
similar	I-Algorithm
to	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Precisely	O
,	O
a	O
matrix	O
A	O
of	O
rank	O
r	O
is	O
an	O
EP	B-Algorithm
matrix	I-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
unitarily	B-Algorithm
similar	I-Algorithm
to	O
a	O
core-nilpotent	O
matrix	O
,	O
that	O
is	O
,	O
</s>
<s>
where	O
U	O
is	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
and	O
C	O
is	O
an	O
r	O
x	O
r	O
nonsingular	O
matrix	O
.	O
</s>
