<s>
In	O
mathematics	O
,	O
trace	B-Application
diagrams	I-Application
are	O
a	O
graphical	O
means	O
of	O
performing	O
computations	O
in	O
linear	B-Language
and	O
multilinear	O
algebra	O
.	O
</s>
<s>
They	O
can	O
be	O
represented	O
as	O
(	O
slightly	O
modified	O
)	O
graphs	O
in	O
which	O
some	O
edges	O
are	O
labeled	O
by	O
matrices	B-Architecture
.	O
</s>
<s>
The	O
simplest	O
trace	B-Application
diagrams	I-Application
represent	O
the	O
trace	O
and	O
determinant	O
of	O
a	O
matrix	O
.	O
</s>
<s>
Several	O
results	O
in	O
linear	B-Language
algebra	I-Language
,	O
such	O
as	O
Cramer	O
's	O
Rule	O
and	O
the	O
Cayley	O
–	O
Hamilton	O
theorem	O
,	O
have	O
simple	O
diagrammatic	O
proofs	O
.	O
</s>
<s>
Let	O
V	O
be	O
a	O
vector	O
space	O
of	O
dimension	O
n	O
over	O
a	O
field	O
F	O
(	O
with	O
n≥2	O
)	O
,	O
and	O
let	O
Hom(V,V )	O
denote	O
the	O
linear	B-Architecture
transformations	I-Architecture
on	O
V	O
.	O
An	O
n-trace	O
diagram	O
is	O
a	O
graph	O
,	O
where	O
the	O
sets	O
Vi	O
(	O
i	O
=	O
1	O
,	O
2	O
,	O
n	O
)	O
are	O
composed	O
of	O
vertices	O
of	O
degree	O
i	O
,	O
together	O
with	O
the	O
following	O
additional	O
structures	O
:	O
</s>
<s>
a	O
labeling	O
V2	O
→	O
Hom(V,V )	O
associating	O
each	O
degree-2	O
vertex	O
to	O
a	O
linear	B-Architecture
transformation	I-Architecture
.	O
</s>
<s>
A	O
framed	O
trace	B-Application
diagram	I-Application
is	O
a	O
trace	B-Application
diagram	I-Application
together	O
with	O
a	O
partition	O
of	O
the	O
degree-1	O
vertices	O
V1	O
into	O
two	O
disjoint	O
ordered	O
collections	O
called	O
the	O
inputs	O
and	O
the	O
outputs	O
.	O
</s>
<s>
The	O
"	O
graph	O
"	O
underlying	O
a	O
trace	B-Application
diagram	I-Application
may	O
have	O
the	O
following	O
special	O
features	O
,	O
which	O
are	O
not	O
always	O
included	O
in	O
the	O
standard	O
definition	O
of	O
a	O
graph	O
:	O
</s>
<s>
When	O
trace	B-Application
diagrams	I-Application
are	O
drawn	O
,	O
the	O
ciliation	O
on	O
an	O
n-vertex	O
is	O
commonly	O
represented	O
by	O
a	O
small	O
mark	O
between	O
two	O
of	O
the	O
incident	O
edges	O
(	O
in	O
the	O
figure	O
above	O
,	O
a	O
small	O
red	O
dot	O
)	O
;	O
the	O
specific	O
ordering	O
of	O
edges	O
follows	O
by	O
proceeding	O
counter-clockwise	O
from	O
this	O
mark	O
.	O
</s>
<s>
Every	O
framed	O
trace	B-Application
diagram	I-Application
corresponds	O
to	O
a	O
multilinear	O
function	O
between	O
tensor	B-Device
powers	O
of	O
the	O
vector	O
space	O
V	O
.	O
The	O
degree-1	O
vertices	O
correspond	O
to	O
the	O
inputs	O
and	O
outputs	O
of	O
the	O
function	O
,	O
while	O
the	O
degree-n	O
vertices	O
correspond	O
to	O
the	O
generalized	O
Levi-Civita	O
symbol	O
(	O
which	O
is	O
an	O
anti-symmetric	O
tensor	B-Device
related	O
to	O
the	O
determinant	O
)	O
.	O
</s>
<s>
If	O
a	O
diagram	O
has	O
no	O
output	O
strands	O
,	O
its	O
function	O
maps	O
tensor	B-Device
products	O
to	O
a	O
scalar	O
.	O
</s>
<s>
By	O
definition	O
,	O
a	O
trace	B-Application
diagram	I-Application
's	O
function	O
is	O
computed	O
using	O
signed	O
graph	O
coloring	O
.	O
</s>
<s>
In	O
practice	O
,	O
a	O
trace	B-Application
diagram	I-Application
's	O
function	O
is	O
typically	O
computed	O
by	O
decomposing	O
the	O
diagram	O
into	O
smaller	O
pieces	O
whose	O
functions	O
are	O
known	O
.	O
</s>
<s>
Several	O
vector	O
identities	O
have	O
easy	O
proofs	O
using	O
trace	B-Application
diagrams	I-Application
.	O
</s>
<s>
Let	O
G	O
be	O
the	O
group	O
of	O
n×n	O
matrices	B-Architecture
.	O
</s>
<s>
If	O
a	O
closed	O
trace	B-Application
diagram	I-Application
is	O
labeled	O
by	O
k	O
different	O
matrices	B-Architecture
,	O
it	O
may	O
be	O
interpreted	O
as	O
a	O
function	O
from	O
to	O
an	O
algebra	O
of	O
multilinear	O
functions	O
.	O
</s>
<s>
Trace	B-Application
diagrams	I-Application
may	O
be	O
specialized	O
for	O
particular	O
Lie	O
groups	O
by	O
altering	O
the	O
definition	O
slightly	O
.	O
</s>
<s>
In	O
this	O
context	O
,	O
they	O
are	O
sometimes	O
called	O
birdtracks	O
,	O
tensor	B-Device
diagrams	O
,	O
or	O
Penrose	O
graphical	O
notation	O
.	O
</s>
<s>
Trace	B-Application
diagrams	I-Application
have	O
primarily	O
been	O
used	O
by	O
physicists	O
as	O
a	O
tool	O
for	O
studying	O
Lie	O
groups	O
.	O
</s>
<s>
The	O
most	O
common	O
applications	O
use	O
representation	O
theory	O
to	O
construct	O
spin	B-Application
networks	I-Application
from	O
trace	B-Application
diagrams	I-Application
.	O
</s>
<s>
In	O
mathematics	O
,	O
they	O
have	O
been	O
used	O
to	O
study	O
character	B-Algorithm
varieties	I-Algorithm
.	O
</s>
