<s>
Equivalent	B-Algorithm
matrices	I-Algorithm
represent	O
the	O
same	O
linear	B-Architecture
transformation	I-Architecture
V	O
→	O
W	O
under	O
two	O
different	O
choices	O
of	O
a	O
pair	O
of	O
bases	O
of	O
V	O
and	O
W	O
,	O
with	O
P	O
and	O
Q	O
being	O
the	O
change	O
of	O
basis	O
matrices	B-Architecture
in	O
V	O
and	O
W	O
respectively	O
.	O
</s>
<s>
The	O
notion	O
of	O
equivalence	O
should	O
not	O
be	O
confused	O
with	O
that	O
of	O
similarity	B-Algorithm
,	O
which	O
is	O
only	O
defined	O
for	O
square	O
matrices	B-Architecture
,	O
and	O
is	O
much	O
more	O
restrictive	O
(	O
similar	B-Algorithm
matrices	I-Algorithm
are	O
certainly	O
equivalent	O
,	O
but	O
equivalent	O
square	O
matrices	B-Architecture
need	O
not	O
be	O
similar	O
)	O
.	O
</s>
<s>
That	O
notion	O
corresponds	O
to	O
matrices	B-Architecture
representing	O
the	O
same	O
endomorphism	O
V	O
→	O
V	O
under	O
two	O
different	O
choices	O
of	O
a	O
single	O
basis	O
of	O
V	O
,	O
used	O
both	O
for	O
initial	O
vectors	O
and	O
their	O
images	O
.	O
</s>
<s>
Matrix	B-Algorithm
equivalence	I-Algorithm
is	O
an	O
equivalence	O
relation	O
on	O
the	O
space	O
of	O
rectangular	O
matrices	B-Architecture
.	O
</s>
<s>
The	O
matrices	B-Architecture
can	O
be	O
transformed	O
into	O
one	O
another	O
by	O
a	O
combination	O
of	O
elementary	O
row	O
and	O
column	O
operations	O
.	O
</s>
<s>
Two	O
matrices	B-Architecture
are	O
equivalent	O
if	O
and	O
only	O
if	O
they	O
have	O
the	O
same	O
rank	O
.	O
</s>
