<s>
In	O
linear	B-Language
algebra	I-Language
,	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
that	O
is	O
equal	O
to	O
its	O
transpose	O
.	O
</s>
<s>
Because	O
equal	O
matrices	O
have	O
equal	O
dimensions	O
,	O
only	O
square	B-Algorithm
matrices	I-Algorithm
can	O
be	O
symmetric	B-Algorithm
.	O
</s>
<s>
The	O
entries	O
of	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
are	O
symmetric	B-Algorithm
with	O
respect	O
to	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
.	O
</s>
<s>
Every	O
square	O
diagonal	B-Algorithm
matrix	I-Algorithm
is	O
symmetric	B-Algorithm
,	O
since	O
all	O
off-diagonal	O
elements	O
are	O
zero	O
.	O
</s>
<s>
Similarly	O
in	O
characteristic	O
different	O
from	O
2	O
,	O
each	O
diagonal	O
element	O
of	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
must	O
be	O
zero	O
,	O
since	O
each	O
is	O
its	O
own	O
negative	O
.	O
</s>
<s>
In	O
linear	B-Language
algebra	I-Language
,	O
a	O
real	O
symmetric	B-Algorithm
matrix	I-Algorithm
represents	O
a	O
self-adjoint	O
operator	O
represented	O
in	O
an	O
orthonormal	O
basis	O
over	O
a	O
real	O
inner	O
product	O
space	O
.	O
</s>
<s>
The	O
corresponding	O
object	O
for	O
a	O
complex	O
inner	O
product	O
space	O
is	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
with	O
complex-valued	O
entries	O
,	O
which	O
is	O
equal	O
to	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
in	O
linear	B-Language
algebra	I-Language
over	O
the	O
complex	O
numbers	O
,	O
it	O
is	O
often	O
assumed	O
that	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
refers	O
to	O
one	O
which	O
has	O
real-valued	O
entries	O
.	O
</s>
<s>
Symmetric	B-Algorithm
matrices	I-Algorithm
appear	O
naturally	O
in	O
a	O
variety	O
of	O
applications	O
,	O
and	O
typical	O
numerical	O
linear	B-Language
algebra	I-Language
software	O
makes	O
special	O
accommodations	O
for	O
them	O
.	O
</s>
<s>
The	O
following	O
matrix	O
is	O
symmetric	B-Algorithm
:	O
</s>
<s>
The	O
sum	O
and	O
difference	O
of	O
two	O
symmetric	B-Algorithm
matrices	I-Algorithm
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
This	O
is	O
not	O
always	O
true	O
for	O
the	O
product	O
:	O
given	O
symmetric	B-Algorithm
matrices	I-Algorithm
and	O
,	O
then	O
is	O
symmetric	B-Algorithm
if	O
and	O
only	O
if	O
and	O
commute	O
,	O
i.e.	O
,	O
if	O
.	O
</s>
<s>
For	O
any	O
integer	O
,	O
is	O
symmetric	B-Algorithm
if	O
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
If	O
exists	O
,	O
it	O
is	O
symmetric	B-Algorithm
if	O
and	O
only	O
if	O
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
Rank	O
of	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
equal	O
to	O
the	O
number	O
of	O
non-zero	O
eigenvalues	O
of	O
.	O
</s>
<s>
Any	O
square	B-Algorithm
matrix	I-Algorithm
can	O
uniquely	O
be	O
written	O
as	O
sum	O
of	O
a	O
symmetric	B-Algorithm
and	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
denotes	O
the	O
space	O
of	O
symmetric	B-Algorithm
matrices	I-Algorithm
and	O
the	O
space	O
of	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
then	O
and	O
,	O
i.e.	O
</s>
<s>
This	O
is	O
true	O
for	O
every	O
square	B-Algorithm
matrix	I-Algorithm
with	O
entries	O
from	O
any	O
field	O
whose	O
characteristic	O
is	O
different	O
from	O
2	O
.	O
</s>
<s>
A	O
symmetric	B-Algorithm
matrix	O
is	O
determined	O
by	O
scalars	O
(	O
the	O
number	O
of	O
entries	O
on	O
or	O
above	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
)	O
.	O
</s>
<s>
Similarly	O
,	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
is	O
determined	O
by	O
scalars	O
(	O
the	O
number	O
of	O
entries	O
above	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
)	O
.	O
</s>
<s>
Any	O
matrix	O
congruent	O
to	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
again	O
symmetric	B-Algorithm
:	O
if	O
is	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
,	O
then	O
so	O
is	O
for	O
any	O
matrix	O
.	O
</s>
<s>
A	O
(	O
real-valued	O
)	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
necessarily	O
a	O
normal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Since	O
this	O
definition	O
is	O
independent	O
of	O
the	O
choice	O
of	O
basis	O
,	O
symmetry	O
is	O
a	O
property	O
that	O
depends	O
only	O
on	O
the	O
linear	B-Architecture
operator	I-Architecture
A	O
and	O
a	O
choice	O
of	O
inner	O
product	O
.	O
</s>
<s>
This	O
characterization	O
of	O
symmetry	O
is	O
useful	O
,	O
for	O
example	O
,	O
in	O
differential	B-Language
geometry	I-Language
,	O
for	O
each	O
tangent	O
space	O
to	O
a	O
manifold	B-Architecture
may	O
be	O
endowed	O
with	O
an	O
inner	O
product	O
,	O
giving	O
rise	O
to	O
what	O
is	O
called	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
.	O
</s>
<s>
The	O
finite-dimensional	O
spectral	O
theorem	O
says	O
that	O
any	O
symmetric	B-Algorithm
matrix	I-Algorithm
whose	O
entries	O
are	O
real	O
can	O
be	O
diagonalized	B-Algorithm
by	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
More	O
explicitly	O
:	O
For	O
every	O
real	O
symmetric	B-Algorithm
matrix	I-Algorithm
there	O
exists	O
a	O
real	B-Algorithm
orthogonal	I-Algorithm
matrix	I-Algorithm
such	O
that	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Every	O
real	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
thus	O
,	O
up	O
to	O
choice	O
of	O
an	O
orthonormal	O
basis	O
,	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
and	O
are	O
real	O
symmetric	B-Algorithm
matrices	I-Algorithm
that	O
commute	O
,	O
then	O
they	O
can	O
be	O
simultaneously	O
diagonalized	B-Algorithm
:	O
there	O
exists	O
a	O
basis	O
of	O
such	O
that	O
every	O
element	O
of	O
the	O
basis	O
is	O
an	O
eigenvector	O
for	O
both	O
and	O
.	O
</s>
<s>
Every	O
real	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
Hermitian	B-Algorithm
,	O
and	O
therefore	O
all	O
its	O
eigenvalues	O
are	O
real	O
.	O
</s>
<s>
(	O
In	O
fact	O
,	O
the	O
eigenvalues	O
are	O
the	O
entries	O
in	O
the	O
diagonal	B-Algorithm
matrix	I-Algorithm
(	O
above	O
)	O
,	O
and	O
therefore	O
is	O
uniquely	O
determined	O
by	O
up	O
to	O
the	O
order	O
of	O
its	O
entries	O
.	O
)	O
</s>
<s>
Essentially	O
,	O
the	O
property	O
of	O
being	O
symmetric	B-Algorithm
for	O
real	O
matrices	O
corresponds	O
to	O
the	O
property	O
of	O
being	O
Hermitian	B-Algorithm
for	O
complex	O
matrices	O
.	O
</s>
<s>
A	O
complex	O
symmetric	B-Algorithm
matrix	I-Algorithm
can	O
be	O
'	O
diagonalized	B-Algorithm
 '	O
using	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
:	O
thus	O
if	O
is	O
a	O
complex	O
symmetric	B-Algorithm
matrix	I-Algorithm
,	O
there	O
is	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
such	O
that	O
is	O
a	O
real	O
diagonal	B-Algorithm
matrix	I-Algorithm
with	O
non-negative	O
entries	O
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
matrix	O
is	O
Hermitian	B-Algorithm
and	O
positive	B-Algorithm
semi-definite	I-Algorithm
,	O
so	O
there	O
is	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
such	O
that	O
is	O
diagonal	O
with	O
non-negative	O
real	O
entries	O
.	O
</s>
<s>
Thus	O
is	O
complex	O
symmetric	B-Algorithm
with	O
real	O
.	O
</s>
<s>
Writing	O
with	O
and	O
real	O
symmetric	B-Algorithm
matrices	I-Algorithm
,	O
.	O
</s>
<s>
Since	O
and	O
commute	O
,	O
there	O
is	O
a	O
real	B-Algorithm
orthogonal	I-Algorithm
matrix	I-Algorithm
such	O
that	O
both	O
and	O
are	O
diagonal	O
.	O
</s>
<s>
Setting	O
(	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
)	O
,	O
the	O
matrix	O
is	O
complex	O
diagonal	O
.	O
</s>
<s>
Pre-multiplying	O
by	O
a	O
suitable	O
diagonal	O
unitary	B-Algorithm
matrix	I-Algorithm
(	O
which	O
preserves	O
unitarity	O
of	O
)	O
,	O
the	O
diagonal	O
entries	O
of	O
can	O
be	O
made	O
to	O
be	O
real	O
and	O
non-negative	O
as	O
desired	O
.	O
</s>
<s>
To	O
construct	O
this	O
matrix	O
,	O
we	O
express	O
the	O
diagonal	B-Algorithm
matrix	I-Algorithm
as	O
.	O
</s>
<s>
(	O
Note	O
,	O
about	O
the	O
eigen-decomposition	O
of	O
a	O
complex	O
symmetric	B-Algorithm
matrix	I-Algorithm
,	O
the	O
Jordan	O
normal	O
form	O
of	O
may	O
not	O
be	O
diagonal	O
,	O
therefore	O
may	O
not	O
be	O
diagonalized	B-Algorithm
by	O
any	O
similarity	O
transformation	O
.	O
)	O
</s>
<s>
Using	O
the	O
Jordan	O
normal	O
form	O
,	O
one	O
can	O
prove	O
that	O
every	O
square	O
real	O
matrix	O
can	O
be	O
written	O
as	O
a	O
product	O
of	O
two	O
real	O
symmetric	B-Algorithm
matrices	I-Algorithm
,	O
and	O
every	O
square	O
complex	O
matrix	O
can	O
be	O
written	O
as	O
a	O
product	O
of	O
two	O
complex	O
symmetric	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Every	O
real	O
non-singular	O
matrix	O
can	O
be	O
uniquely	O
factored	O
as	O
the	O
product	O
of	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
and	O
a	O
symmetric	B-Algorithm
positive	I-Algorithm
definite	I-Algorithm
matrix	O
,	O
which	O
is	O
called	O
a	O
polar	O
decomposition	O
.	O
</s>
<s>
A	O
general	O
(	O
complex	O
)	O
symmetric	B-Algorithm
matrix	I-Algorithm
may	O
be	O
defective	O
and	O
thus	O
not	O
be	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
where	O
is	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
,	O
and	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
of	O
the	O
eigenvalues	O
of	O
.	O
</s>
<s>
In	O
the	O
special	O
case	O
that	O
is	O
real	O
symmetric	B-Algorithm
,	O
then	O
and	O
are	O
also	O
real	O
.	O
</s>
<s>
Symmetric	B-Algorithm
matrices	O
of	O
real	O
functions	O
appear	O
as	O
the	O
Hessians	O
of	O
twice	O
differentiable	O
functions	O
of	O
real	O
variables	O
(	O
the	O
continuity	O
of	O
the	O
second	O
derivative	O
is	O
not	O
needed	O
,	O
despite	O
common	O
belief	O
to	O
the	O
opposite	O
)	O
.	O
</s>
<s>
Every	O
quadratic	O
form	O
on	O
can	O
be	O
uniquely	O
written	O
in	O
the	O
form	O
with	O
a	O
symmetric	B-Algorithm
matrix	O
.	O
</s>
<s>
The	O
transpose	O
of	O
a	O
symmetrizable	B-Algorithm
matrix	I-Algorithm
is	O
symmetrizable	O
,	O
since	O
and	O
is	O
symmetric	B-Algorithm
.	O
</s>
