<s>
In	O
linear	O
algebra	O
,	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
of	O
size	O
is	O
the	O
square	B-Algorithm
matrix	I-Algorithm
with	O
ones	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
and	O
zeros	O
elsewhere	O
.	O
</s>
<s>
The	O
identity	B-Algorithm
matrix	I-Algorithm
is	O
often	O
denoted	O
by	O
,	O
or	O
simply	O
by	O
if	O
the	O
size	O
is	O
immaterial	O
or	O
can	O
be	O
trivially	O
determined	O
by	O
the	O
context	O
.	O
</s>
<s>
The	O
term	O
unit	B-Algorithm
matrix	I-Algorithm
has	O
also	O
been	O
widely	O
used	O
,	O
but	O
the	O
term	O
identity	B-Algorithm
matrix	I-Algorithm
is	O
now	O
standard	O
.	O
</s>
<s>
The	O
term	O
unit	B-Algorithm
matrix	I-Algorithm
is	O
ambiguous	O
,	O
because	O
it	O
is	O
also	O
used	O
for	O
a	O
matrix	O
of	O
ones	O
and	O
for	O
any	O
unit	O
of	O
the	O
ring	O
of	O
all	O
matrices	O
.	O
</s>
<s>
In	O
some	O
fields	O
,	O
such	O
as	O
group	O
theory	O
or	O
quantum	O
mechanics	O
,	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
is	O
sometimes	O
denoted	O
by	O
a	O
boldface	O
one	O
,	O
,	O
or	O
called	O
"	O
id	O
"	O
(	O
short	O
for	O
identity	O
)	O
.	O
</s>
<s>
Less	O
frequently	O
,	O
some	O
mathematics	O
books	O
use	O
or	O
to	O
represent	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
standing	O
for	O
"	O
unit	B-Algorithm
matrix	I-Algorithm
"	O
and	O
the	O
German	O
word	O
respectively	O
.	O
</s>
<s>
The	O
identity	B-Algorithm
matrix	I-Algorithm
can	O
also	O
be	O
written	O
using	O
the	O
Kronecker	O
delta	O
notation	O
:	O
</s>
<s>
In	O
particular	O
,	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
serves	O
as	O
the	O
multiplicative	O
identity	O
of	O
the	O
matrix	O
ring	O
of	O
all	O
matrices	O
,	O
and	O
as	O
the	O
identity	O
element	O
of	O
the	O
general	O
linear	O
group	O
,	O
which	O
consists	O
of	O
all	O
invertible	O
matrices	O
under	O
the	O
matrix	O
multiplication	O
operation	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
is	O
invertible	O
.	O
</s>
<s>
It	O
is	O
an	O
involutory	B-Algorithm
matrix	I-Algorithm
,	O
equal	O
to	O
its	O
own	O
inverse	O
.	O
</s>
<s>
In	O
this	O
group	O
,	O
two	O
square	B-Algorithm
matrices	I-Algorithm
have	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
as	O
their	O
product	O
exactly	O
when	O
they	O
are	O
the	O
inverses	O
of	O
each	O
other	O
.	O
</s>
<s>
When	O
matrices	O
are	O
used	O
to	O
represent	O
linear	B-Architecture
transformations	I-Architecture
from	O
an	O
-dimensional	O
vector	O
space	O
to	O
itself	O
,	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
represents	O
the	O
identity	O
function	O
,	O
for	O
whatever	O
basis	O
was	O
used	O
in	O
this	O
representation	O
.	O
</s>
<s>
The	O
th	O
column	O
of	O
an	O
identity	B-Algorithm
matrix	I-Algorithm
is	O
the	O
unit	O
vector	O
,	O
a	O
vector	O
whose	O
th	O
entry	O
is	O
1	O
and	O
0	O
elsewhere	O
.	O
</s>
<s>
The	O
determinant	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
is1	O
,	O
and	O
its	O
trace	O
is	O
.	O
</s>
<s>
The	O
identity	B-Algorithm
matrix	I-Algorithm
is	O
the	O
only	O
idempotent	O
matrix	O
with	O
non-zero	O
determinant	O
.	O
</s>
<s>
The	O
principal	O
square	O
root	O
of	O
an	O
identity	B-Algorithm
matrix	I-Algorithm
is	O
itself	O
,	O
and	O
this	O
is	O
its	O
only	O
positive-definite	B-Algorithm
square	O
root	O
.	O
</s>
<s>
However	O
,	O
every	O
identity	B-Algorithm
matrix	I-Algorithm
with	O
at	O
least	O
two	O
rows	O
and	O
columns	O
has	O
an	O
infinitude	O
of	O
symmetric	O
square	O
roots	O
.	O
</s>
<s>
The	O
rank	O
of	O
an	O
identity	B-Algorithm
matrix	I-Algorithm
equals	O
the	O
size	O
,	O
i.e.	O
</s>
