<s>
Similar	B-Algorithm
matrices	I-Algorithm
represent	O
the	O
same	O
linear	B-Architecture
map	I-Architecture
under	O
two	O
(	O
possibly	O
)	O
different	O
bases	O
,	O
with	O
being	O
the	O
change	O
of	O
basis	O
matrix	O
.	O
</s>
<s>
In	O
the	O
general	O
linear	O
group	O
,	O
similarity	O
is	O
therefore	O
the	O
same	O
as	O
conjugacy	O
,	O
and	O
similar	B-Algorithm
matrices	I-Algorithm
are	O
also	O
called	O
conjugate	O
;	O
however	O
,	O
in	O
a	O
given	O
subgroup	O
of	O
the	O
general	O
linear	O
group	O
,	O
the	O
notion	O
of	O
conjugacy	O
may	O
be	O
more	O
restrictive	O
than	O
similarity	O
,	O
since	O
it	O
requires	O
that	O
be	O
chosen	O
to	O
lie	O
in	O
.	O
</s>
<s>
When	O
defining	O
a	O
linear	B-Architecture
transformation	I-Architecture
,	O
it	O
can	O
be	O
the	O
case	O
that	O
a	O
change	O
of	O
basis	O
can	O
result	O
in	O
a	O
simpler	O
form	O
of	O
the	O
same	O
transformation	O
.	O
</s>
<s>
The	O
transform	O
in	O
the	O
original	O
basis	O
is	O
found	O
to	O
be	O
the	O
product	O
of	O
three	O
easy-to-derive	O
matrices	B-Architecture
.	O
</s>
<s>
Similarity	O
is	O
an	O
equivalence	O
relation	O
on	O
the	O
space	O
of	O
square	O
matrices	B-Architecture
.	O
</s>
<s>
Because	O
matrices	B-Architecture
are	O
similar	O
if	O
and	O
only	O
if	O
they	O
represent	O
the	O
same	O
linear	B-Architecture
operator	I-Architecture
with	O
respect	O
to	O
(	O
possibly	O
)	O
different	O
bases	O
,	O
similar	B-Algorithm
matrices	I-Algorithm
share	O
all	O
properties	O
of	O
their	O
shared	O
underlying	O
operator	O
:	O
</s>
<s>
For	O
example	O
,	O
A	O
is	O
called	O
diagonalizable	B-Algorithm
if	O
it	O
is	O
similar	O
to	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Not	O
all	O
matrices	B-Architecture
are	O
diagonalizable	B-Algorithm
,	O
but	O
at	O
least	O
over	O
the	O
complex	O
numbers	O
(	O
or	O
any	O
algebraically	O
closed	O
field	O
)	O
,	O
every	O
matrix	O
is	O
similar	O
to	O
a	O
matrix	O
in	O
Jordan	O
form	O
.	O
</s>
<s>
Note	O
that	O
this	O
Smith	O
normal	O
form	O
is	O
not	O
a	O
normal	O
form	O
of	O
A	O
itself	O
;	O
moreover	O
it	O
is	O
not	O
similar	O
to	O
either	O
,	O
but	O
obtained	O
from	O
the	O
latter	O
by	O
left	O
and	O
right	O
multiplications	O
by	O
different	O
invertible	O
matrices	B-Architecture
(	O
with	O
polynomial	O
entries	O
)	O
.	O
</s>
<s>
Similarity	O
of	O
matrices	B-Architecture
does	O
not	O
depend	O
on	O
the	O
base	O
field	O
:	O
if	O
L	O
is	O
a	O
field	O
containing	O
K	O
as	O
a	O
subfield	O
,	O
and	O
A	O
and	O
B	O
are	O
two	O
matrices	B-Architecture
over	O
K	O
,	O
then	O
A	O
and	O
B	O
are	O
similar	O
as	O
matrices	B-Architecture
over	O
K	O
if	O
and	O
only	O
if	O
they	O
are	O
similar	O
as	O
matrices	B-Architecture
over	O
L	O
.	O
This	O
is	O
so	O
because	O
the	O
rational	O
canonical	O
form	O
over	O
K	O
is	O
also	O
the	O
rational	O
canonical	O
form	O
over	O
L	O
.	O
This	O
means	O
that	O
one	O
may	O
use	O
Jordan	O
forms	O
that	O
only	O
exist	O
over	O
a	O
larger	O
field	O
to	O
determine	O
whether	O
the	O
given	O
matrices	B-Architecture
are	O
similar	O
.	O
</s>
<s>
In	O
the	O
definition	O
of	O
similarity	O
,	O
if	O
the	O
matrix	O
P	O
can	O
be	O
chosen	O
to	O
be	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
then	O
A	O
and	O
B	O
are	O
permutation-similar	O
;	O
if	O
P	O
can	O
be	O
chosen	O
to	O
be	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
then	O
A	O
and	O
B	O
are	O
unitarily	O
equivalent	O
.	O
</s>
<s>
The	O
spectral	O
theorem	O
says	O
that	O
every	O
normal	B-Algorithm
matrix	I-Algorithm
is	O
unitarily	O
equivalent	O
to	O
some	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Specht	O
's	O
theorem	O
states	O
that	O
two	O
matrices	B-Architecture
are	O
unitarily	O
equivalent	O
if	O
and	O
only	O
if	O
they	O
satisfy	O
certain	O
trace	O
equalities	O
.	O
</s>
