<s>
In	O
mathematics	O
,	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
(	O
or	O
self-adjoint	O
matrix	O
)	O
is	O
a	O
complex	O
square	B-Algorithm
matrix	I-Algorithm
that	O
is	O
equal	O
to	O
its	O
own	O
conjugate	B-Algorithm
transpose	I-Algorithm
—	O
that	O
is	O
,	O
the	O
element	O
in	O
the	O
-th	O
row	O
and	O
-th	O
column	O
is	O
equal	O
to	O
the	O
complex	O
conjugate	O
of	O
the	O
element	O
in	O
the	O
-th	O
row	O
and	O
-th	O
column	O
,	O
for	O
all	O
indices	O
and	O
:	O
</s>
<s>
Hermitian	B-Algorithm
matrices	I-Algorithm
can	O
be	O
understood	O
as	O
the	O
complex	O
extension	O
of	O
real	O
symmetric	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Hermitian	B-Algorithm
matrices	I-Algorithm
are	O
named	O
after	O
Charles	O
Hermite	O
,	O
who	O
demonstrated	O
in	O
1855	O
that	O
matrices	O
of	O
this	O
form	O
share	O
a	O
property	O
with	O
real	O
symmetric	B-Algorithm
matrices	I-Algorithm
of	O
always	O
having	O
real	O
eigenvalues	O
.	O
</s>
<s>
Other	O
,	O
equivalent	O
notations	O
in	O
common	O
use	O
are	O
although	O
in	O
quantum	O
mechanics	O
,	O
typically	O
means	O
the	O
complex	O
conjugate	O
only	O
,	O
and	O
not	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
.	O
</s>
<s>
Hermitian	B-Algorithm
matrices	I-Algorithm
can	O
be	O
characterized	O
in	O
a	O
number	O
of	O
equivalent	O
ways	O
,	O
some	O
of	O
which	O
are	O
listed	O
below	O
:	O
</s>
<s>
A	O
square	B-Algorithm
matrix	I-Algorithm
is	O
Hermitian	O
if	O
and	O
only	O
if	O
it	O
is	O
unitarily	O
diagonalizable	B-Algorithm
with	O
real	O
eigenvalues	O
.	O
</s>
<s>
Hermitian	B-Algorithm
matrices	I-Algorithm
are	O
fundamental	O
to	O
quantum	O
mechanics	O
because	O
they	O
describe	O
operators	O
with	O
necessarily	O
real	O
eigenvalues	O
.	O
</s>
<s>
Well-known	O
families	O
of	O
Hermitian	B-Algorithm
matrices	I-Algorithm
include	O
the	O
Pauli	O
matrices	O
,	O
the	O
Gell-Mann	O
matrices	O
and	O
their	O
generalizations	O
.	O
</s>
<s>
In	O
theoretical	O
physics	O
such	O
Hermitian	B-Algorithm
matrices	I-Algorithm
are	O
often	O
multiplied	O
by	O
imaginary	O
coefficients	O
,	O
Physics	O
125	O
Course	O
Notes	O
at	O
California	O
Institute	O
of	O
Technology	O
which	O
results	O
in	O
skew-Hermitian	O
matrices	O
.	O
</s>
<s>
Here	O
,	O
we	O
offer	O
another	O
useful	O
Hermitian	B-Algorithm
matrix	I-Algorithm
using	O
an	O
abstract	O
example	O
.	O
</s>
<s>
If	O
a	O
square	B-Algorithm
matrix	I-Algorithm
equals	O
the	O
product	O
of	O
a	O
matrix	O
with	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
,	O
that	O
is	O
,	O
then	O
is	O
a	O
Hermitian	O
positive	B-Algorithm
semi-definite	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
entries	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
(	O
top	O
left	O
to	O
bottom	O
right	O
)	O
of	O
any	O
Hermitian	B-Algorithm
matrix	I-Algorithm
are	O
real	O
.	O
</s>
<s>
Only	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
entries	O
are	O
necessarily	O
real	O
;	O
Hermitian	B-Algorithm
matrices	I-Algorithm
can	O
have	O
arbitrary	O
complex-valued	O
entries	O
in	O
their	O
off-diagonal	O
elements	O
,	O
as	O
long	O
as	O
diagonally-opposite	O
entries	O
are	O
complex	O
conjugates	O
.	O
</s>
<s>
A	O
matrix	O
that	O
has	O
only	O
real	O
entries	O
is	O
symmetric	B-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
Hermitian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
real	O
and	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
simply	O
a	O
special	O
case	O
of	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Every	O
Hermitian	B-Algorithm
matrix	I-Algorithm
is	O
a	O
normal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
finite-dimensional	O
spectral	O
theorem	O
says	O
that	O
any	O
Hermitian	B-Algorithm
matrix	I-Algorithm
can	O
be	O
diagonalized	B-Algorithm
by	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
,	O
and	O
that	O
the	O
resulting	O
diagonal	O
matrix	O
has	O
only	O
real	O
entries	O
.	O
</s>
<s>
This	O
implies	O
that	O
all	O
eigenvalues	O
of	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
with	O
dimension	O
are	O
real	O
,	O
and	O
that	O
has	O
linearly	O
independent	O
eigenvectors	O
.	O
</s>
<s>
Moreover	O
,	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
has	O
orthogonal	O
eigenvectors	O
for	O
distinct	O
eigenvalues	O
.	O
</s>
<s>
Even	O
if	O
there	O
are	O
degenerate	O
eigenvalues	O
,	O
it	O
is	O
always	O
possible	O
to	O
find	O
an	O
orthogonal	B-Algorithm
basis	I-Algorithm
of	O
consisting	O
of	O
eigenvectors	O
of	O
.	O
</s>
<s>
The	O
sum	O
of	O
any	O
two	O
Hermitian	B-Algorithm
matrices	I-Algorithm
is	O
Hermitian	O
.	O
</s>
<s>
The	O
inverse	O
of	O
an	O
invertible	O
Hermitian	B-Algorithm
matrix	I-Algorithm
is	O
Hermitian	O
as	O
well	O
.	O
</s>
<s>
The	O
product	O
of	O
two	O
Hermitian	B-Algorithm
matrices	I-Algorithm
and	O
is	O
Hermitian	O
if	O
and	O
only	O
if	O
.	O
</s>
<s>
For	O
an	O
arbitrary	O
complex	O
valued	O
vector	O
the	O
product	O
is	O
real	O
because	O
of	O
This	O
is	O
especially	O
important	O
in	O
quantum	O
physics	O
where	O
Hermitian	B-Algorithm
matrices	I-Algorithm
are	O
operators	O
that	O
measure	O
properties	O
of	O
a	O
system	O
,	O
e.g.	O
</s>
<s>
However	O
the	O
complex	O
Hermitian	B-Algorithm
matrices	I-Algorithm
do''	O
form	O
a	O
vector	O
space	O
over	O
the	O
real	O
numbers	O
.	O
</s>
<s>
In	O
the	O
-dimensional	O
vector	O
space	O
of	O
complex	O
matrices	O
over	O
,	O
the	O
complex	O
Hermitian	B-Algorithm
matrices	I-Algorithm
form	O
a	O
subspace	O
of	O
dimension	O
.	O
</s>
<s>
An	O
example	O
is	O
that	O
the	O
four	O
Pauli	O
matrices	O
form	O
a	O
complete	O
basis	O
for	O
the	O
vector	O
space	O
of	O
all	O
complex	O
2-by-2	O
Hermitian	B-Algorithm
matrices	I-Algorithm
over	O
.	O
</s>
<s>
Since	O
has	O
an	O
eigendecomposition	O
,	O
where	O
is	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
(	O
its	O
columns	O
are	O
orthonormal	O
vectors	O
;	O
see	O
above	O
)	O
,	O
a	O
singular	O
value	O
decomposition	O
of	O
is	O
,	O
where	O
and	O
are	O
diagonal	O
matrices	O
containing	O
the	O
absolute	O
values	O
and	O
signs	O
of	O
'	O
s	O
eigenvalues	O
,	O
respectively	O
.	O
</s>
<s>
The	O
determinant	O
of	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
is	O
real	O
:	O
</s>
<s>
(	O
Alternatively	O
,	O
the	O
determinant	O
is	O
the	O
product	O
of	O
the	O
matrix	O
's	O
eigenvalues	O
,	O
and	O
as	O
mentioned	O
before	O
,	O
the	O
eigenvalues	O
of	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
are	O
real	O
.	O
)	O
</s>
<s>
Additional	O
facts	O
related	O
to	O
Hermitian	B-Algorithm
matrices	I-Algorithm
include	O
:	O
</s>
<s>
The	O
sum	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
and	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
is	O
Hermitian	O
.	O
</s>
<s>
The	O
difference	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
and	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
is	O
skew-Hermitian	O
(	O
also	O
called	O
antihermitian	O
)	O
.	O
</s>
<s>
This	O
implies	O
that	O
the	O
commutator	O
of	O
two	O
Hermitian	B-Algorithm
matrices	I-Algorithm
is	O
skew-Hermitian	O
.	O
</s>
<s>
An	O
arbitrary	O
square	B-Algorithm
matrix	I-Algorithm
can	O
be	O
written	O
as	O
the	O
sum	O
of	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
and	O
a	O
skew-Hermitian	O
matrix	O
.	O
</s>
<s>
In	O
mathematics	O
,	O
for	O
a	O
given	O
complex	O
Hermitian	B-Algorithm
matrix	I-Algorithm
and	O
nonzero	O
vector	O
,	O
the	O
Rayleigh	O
quotient	O
is	O
defined	O
as	O
:	O
</s>
<s>
For	O
real	O
matrices	O
and	O
vectors	O
,	O
the	O
condition	O
of	O
being	O
Hermitian	O
reduces	O
to	O
that	O
of	O
being	O
symmetric	B-Algorithm
,	O
and	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
to	O
the	O
usual	O
transpose	O
for	O
any	O
non-zero	O
real	O
scalar	O
Also	O
,	O
recall	O
that	O
a	O
Hermitian	O
(	O
or	O
real	O
symmetric	B-Algorithm
)	O
matrix	O
has	O
real	O
eigenvalues	O
.	O
</s>
