<s>
In	O
mathematics	O
,	O
a	O
complex	O
square	B-Algorithm
matrix	I-Algorithm
is	O
normal	O
if	O
it	O
commutes	O
with	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
:	O
</s>
<s>
The	O
concept	O
of	O
normal	B-Algorithm
matrices	I-Algorithm
can	O
be	O
extended	O
to	O
normal	O
operators	O
on	O
infinite	O
dimensional	O
normed	O
spaces	O
and	O
to	O
normal	O
elements	O
in	O
C*	B-Algorithm
-algebras	I-Algorithm
.	O
</s>
<s>
This	O
makes	O
normal	O
operators	O
,	O
and	O
normal	O
elements	O
of	O
C*	B-Algorithm
-algebras	I-Algorithm
,	O
more	O
amenable	O
to	O
analysis	O
.	O
</s>
<s>
The	O
spectral	O
theorem	O
states	O
that	O
a	O
matrix	O
is	O
normal	O
if	O
and	O
only	O
if	O
it	O
is	O
unitarily	B-Algorithm
similar	I-Algorithm
to	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
,	O
and	O
therefore	O
any	O
matrix	O
satisfying	O
the	O
equation	O
is	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
The	O
converse	O
does	O
not	O
hold	O
because	O
diagonalizable	B-Algorithm
matrices	I-Algorithm
may	O
have	O
non-orthogonal	O
eigenspaces	O
.	O
</s>
<s>
The	O
left	O
and	O
right	O
singular	O
vectors	O
in	O
the	O
singular	O
value	O
decomposition	O
of	O
a	O
normal	B-Algorithm
matrix	I-Algorithm
differ	O
only	O
in	O
complex	O
phase	O
from	O
each	O
other	O
and	O
from	O
the	O
corresponding	O
eigenvectors	O
,	O
since	O
the	O
phase	O
must	O
be	O
factored	O
out	O
of	O
the	O
eigenvalues	O
to	O
form	O
singular	O
values	O
.	O
</s>
<s>
Among	O
complex	O
matrices	O
,	O
all	O
unitary	B-Algorithm
,	O
Hermitian	B-Algorithm
,	O
and	O
skew-Hermitian	O
matrices	O
are	O
normal	O
,	O
with	O
all	O
eigenvalues	O
being	O
unit	O
modulus	O
,	O
real	O
,	O
and	O
imaginary	O
,	O
respectively	O
.	O
</s>
<s>
Likewise	O
,	O
among	O
real	O
matrices	O
,	O
all	O
orthogonal	B-Algorithm
,	O
symmetric	B-Algorithm
,	O
and	O
skew-symmetric	B-Algorithm
matrices	I-Algorithm
are	O
normal	O
,	O
with	O
all	O
eigenvalues	O
being	O
complex	O
conjugate	O
pairs	O
on	O
the	O
unit	O
circle	O
,	O
real	O
,	O
and	O
imaginary	O
,	O
respectively	O
.	O
</s>
<s>
However	O
,	O
it	O
is	O
not	O
the	O
case	O
that	O
all	O
normal	B-Algorithm
matrices	I-Algorithm
are	O
either	O
unitary	B-Algorithm
or	O
(	O
skew	O
-	O
)	O
Hermitian	B-Algorithm
,	O
as	O
their	O
eigenvalues	O
can	O
be	O
any	O
complex	O
number	O
,	O
in	O
general	O
.	O
</s>
<s>
The	O
concept	O
of	O
normality	O
is	O
important	O
because	O
normal	B-Algorithm
matrices	I-Algorithm
are	O
precisely	O
those	O
to	O
which	O
the	O
spectral	O
theorem	O
applies	O
:	O
</s>
<s>
The	O
diagonal	B-Algorithm
entries	O
of	O
are	O
the	O
eigenvalues	O
of	O
,	O
and	O
the	O
columns	O
of	O
are	O
the	O
eigenvectors	O
of	O
.	O
</s>
<s>
Another	O
way	O
of	O
stating	O
the	O
spectral	O
theorem	O
is	O
to	O
say	O
that	O
normal	B-Algorithm
matrices	I-Algorithm
are	O
precisely	O
those	O
matrices	O
that	O
can	O
be	O
represented	O
by	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
with	O
respect	O
to	O
a	O
properly	O
chosen	O
orthonormal	O
basis	O
of	O
.	O
</s>
<s>
Phrased	O
differently	O
:	O
a	O
matrix	O
is	O
normal	O
if	O
and	O
only	O
if	O
its	O
eigenspaces	O
span	O
and	O
are	O
pairwise	O
orthogonal	B-Algorithm
with	O
respect	O
to	O
the	O
standard	O
inner	O
product	O
of	O
.	O
</s>
<s>
The	O
spectral	O
theorem	O
for	O
normal	B-Algorithm
matrices	I-Algorithm
is	O
a	O
special	O
case	O
of	O
the	O
more	O
general	O
Schur	O
decomposition	O
which	O
holds	O
for	O
all	O
square	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Let	O
be	O
a	O
square	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Then	O
by	O
Schur	O
decomposition	O
it	O
is	O
unitary	B-Algorithm
similar	O
to	O
an	O
upper-triangular	B-Algorithm
matrix	I-Algorithm
,	O
say	O
,	O
.	O
</s>
<s>
But	O
then	O
must	O
be	O
diagonal	B-Algorithm
,	O
for	O
,	O
as	O
noted	O
above	O
,	O
a	O
normal	O
upper-triangular	B-Algorithm
matrix	I-Algorithm
is	O
diagonal	B-Algorithm
.	O
</s>
<s>
The	O
spectral	O
theorem	O
permits	O
the	O
classification	O
of	O
normal	B-Algorithm
matrices	I-Algorithm
in	O
terms	O
of	O
their	O
spectra	O
,	O
for	O
example	O
:	O
</s>
<s>
In	O
general	O
,	O
the	O
sum	O
or	O
product	O
of	O
two	O
normal	B-Algorithm
matrices	I-Algorithm
need	O
not	O
be	O
normal	O
.	O
</s>
<s>
This	O
follows	O
by	O
combining	O
the	O
theorems	O
that	O
,	O
over	O
an	O
algebraically	O
closed	O
field	O
,	O
commuting	O
matrices	O
are	O
simultaneously	O
triangularizable	B-Algorithm
and	O
a	O
normal	B-Algorithm
matrix	I-Algorithm
is	O
diagonalizable	B-Algorithm
–	O
the	O
added	O
result	O
is	O
that	O
these	O
can	O
both	O
be	O
done	O
simultaneously	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
give	O
a	O
fairly	O
long	O
list	O
of	O
equivalent	O
definitions	O
of	O
a	O
normal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
is	O
diagonalizable	B-Algorithm
by	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
Hermitian	B-Algorithm
part	O
and	O
skew-Hermitian	O
part	O
of	O
commute	O
.	O
</s>
<s>
for	O
some	O
unitary	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
and	O
commute	O
,	O
where	O
we	O
have	O
the	O
polar	O
decomposition	O
with	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
and	O
some	O
positive	B-Algorithm
semidefinite	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
commutes	O
with	O
some	O
normal	B-Algorithm
matrix	I-Algorithm
with	O
distinct	O
eigenvalues	O
.	O
</s>
<s>
It	O
is	O
occasionally	O
useful	O
(	O
but	O
sometimes	O
misleading	O
)	O
to	O
think	O
of	O
the	O
relationships	O
of	O
special	O
kinds	O
of	O
normal	B-Algorithm
matrices	I-Algorithm
as	O
analogous	O
to	O
the	O
relationships	O
of	O
the	O
corresponding	O
type	O
of	O
complex	O
numbers	O
of	O
which	O
their	O
eigenvalues	O
are	O
composed	O
.	O
</s>
<s>
This	O
is	O
because	O
any	O
function	O
of	O
a	O
non-defective	O
matrix	O
acts	O
directly	O
on	O
each	O
of	O
its	O
eigenvalues	O
,	O
and	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
its	O
spectral	O
decomposition	O
is	O
,	O
where	O
is	O
the	O
diagonal	B-Algorithm
matrix	I-Algorithm
of	O
eigenvalues	O
.	O
</s>
<s>
Likewise	O
,	O
if	O
two	O
normal	B-Algorithm
matrices	I-Algorithm
commute	O
and	O
are	O
therefore	O
simultaneously	O
diagonalizable	B-Algorithm
,	O
any	O
operation	O
between	O
these	O
matrices	O
also	O
acts	O
on	O
each	O
corresponding	O
pair	O
of	O
eigenvalues	O
.	O
</s>
<s>
The	O
conjugate	B-Algorithm
transpose	I-Algorithm
is	O
analogous	O
to	O
the	O
complex	O
conjugate	O
.	O
</s>
<s>
Unitary	B-Algorithm
matrices	I-Algorithm
are	O
analogous	O
to	O
complex	O
numbers	O
on	O
the	O
unit	O
circle	O
.	O
</s>
<s>
Hermitian	B-Algorithm
matrices	I-Algorithm
are	O
analogous	O
to	O
real	O
numbers	O
.	O
</s>
<s>
Hermitian	B-Algorithm
positive	B-Algorithm
definite	I-Algorithm
matrices	I-Algorithm
are	O
analogous	O
to	O
positive	O
real	O
numbers	O
.	O
</s>
<s>
Skew	O
Hermitian	B-Algorithm
matrices	I-Algorithm
are	O
analogous	O
to	O
purely	O
imaginary	O
numbers	O
.	O
</s>
<s>
Invertible	O
matrices	O
are	O
analogous	O
to	O
non-zero	O
complex	O
numbers	O
.	O
</s>
<s>
the	O
identity	B-Algorithm
matrix	I-Algorithm
is	O
analogous	O
to	O
1	O
.	O
</s>
<s>
An	O
idempotent	O
matrix	O
is	O
an	O
orthogonal	B-Algorithm
projection	O
with	O
each	O
eigenvalue	O
either	O
0	O
or	O
1	O
.	O
</s>
