<s>
In	O
linear	B-Language
algebra	I-Language
,	O
a	O
square	B-Algorithm
matrix	I-Algorithm
is	O
called	O
diagonalizable	B-Algorithm
or	O
non-defective	O
if	O
it	O
is	O
similar	B-Algorithm
to	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
,	O
i.e.	O
,	O
if	O
there	O
exists	O
an	O
invertible	O
matrix	B-Architecture
and	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
such	O
that	O
or	O
equivalently	O
(	O
Such	O
are	O
not	O
unique	O
.	O
)	O
</s>
<s>
For	O
a	O
finite-dimensional	O
vector	O
space	O
a	O
linear	B-Architecture
map	I-Architecture
is	O
called	O
diagonalizable	B-Algorithm
if	O
there	O
exists	O
an	O
ordered	O
basis	O
of	O
consisting	O
of	O
eigenvectors	O
of	O
.	O
</s>
<s>
Diagonalizable	B-Algorithm
matrices	I-Algorithm
and	O
maps	O
are	O
especially	O
easy	O
for	O
computations	O
,	O
once	O
their	O
eigenvalues	O
and	O
eigenvectors	O
are	O
known	O
.	O
</s>
<s>
One	O
can	O
raise	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
to	O
a	O
power	O
by	O
simply	O
raising	O
the	O
diagonal	O
entries	O
to	O
that	O
power	O
,	O
and	O
the	O
determinant	O
of	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
is	O
simply	O
the	O
product	O
of	O
all	O
diagonal	O
entries	O
;	O
such	O
computations	O
generalize	O
easily	O
to	O
Geometrically	O
,	O
a	O
diagonalizable	B-Algorithm
matrix	I-Algorithm
is	O
an	O
inhomogeneous	B-Algorithm
dilation	I-Algorithm
(	O
or	O
anisotropic	O
scaling	B-Algorithm
)	O
—	O
it	O
scales	B-Algorithm
the	O
space	O
,	O
as	O
does	O
a	O
homogeneous	B-Algorithm
dilation	I-Algorithm
,	O
but	O
by	O
a	O
different	O
factor	O
along	O
each	O
eigenvector	O
axis	O
,	O
the	O
factor	O
given	O
by	O
the	O
corresponding	O
eigenvalue	O
.	O
</s>
<s>
A	O
square	B-Algorithm
matrix	I-Algorithm
that	O
is	O
not	O
diagonalizable	B-Algorithm
is	O
called	O
defective	O
.	O
</s>
<s>
It	O
can	O
happen	O
that	O
a	O
matrix	B-Architecture
with	O
real	O
entries	O
is	O
defective	O
over	O
the	O
real	O
numbers	O
,	O
meaning	O
that	O
is	O
impossible	O
for	O
any	O
invertible	O
and	O
diagonal	O
with	O
real	O
entries	O
,	O
but	O
it	O
is	O
possible	O
with	O
complex	O
entries	O
,	O
so	O
that	O
is	O
diagonalizable	B-Algorithm
over	O
the	O
complex	O
numbers	O
.	O
</s>
<s>
For	O
example	O
,	O
this	O
is	O
the	O
case	O
for	O
a	O
generic	O
rotation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Many	O
results	O
for	O
diagonalizable	B-Algorithm
matrices	I-Algorithm
hold	O
only	O
over	O
an	O
algebraically	O
closed	O
field	O
(	O
such	O
as	O
the	O
complex	O
numbers	O
)	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
diagonalizable	B-Algorithm
matrices	I-Algorithm
are	O
dense	O
in	O
the	O
space	O
of	O
all	O
matrices	O
,	O
which	O
means	O
any	O
defective	O
matrix	B-Architecture
can	O
be	O
deformed	O
into	O
a	O
diagonalizable	B-Algorithm
matrix	I-Algorithm
by	O
a	O
small	O
perturbation	O
;	O
and	O
the	O
Jordan	O
normal	B-Algorithm
form	O
theorem	O
states	O
that	O
any	O
matrix	B-Architecture
is	O
uniquely	O
the	O
sum	O
of	O
a	O
diagonalizable	B-Algorithm
matrix	I-Algorithm
and	O
a	O
nilpotent	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Over	O
an	O
algebraically	O
closed	O
field	O
,	O
diagonalizable	B-Algorithm
matrices	I-Algorithm
are	O
equivalent	O
to	O
semi-simple	O
matrices	O
.	O
</s>
<s>
A	O
square	O
matrix	B-Architecture
,	O
,	O
with	O
entries	O
in	O
a	O
field	O
is	O
called	O
diagonalizable	B-Algorithm
or	O
nondefective	O
if	O
there	O
exists	O
an	O
invertible	O
matrix	B-Architecture
(	O
i.e.	O
</s>
<s>
an	O
element	O
of	O
the	O
general	O
linear	O
group	O
GLn(F )	O
)	O
,	O
,	O
such	O
that	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
fundamental	O
fact	O
about	O
diagonalizable	B-Algorithm
maps	O
and	O
matrices	O
is	O
expressed	O
by	O
the	O
following	O
:	O
</s>
<s>
An	O
matrix	B-Architecture
over	O
a	O
field	O
is	O
diagonalizable	B-Algorithm
if	O
and	O
only	O
if	O
the	O
sum	O
of	O
the	O
dimensions	O
of	O
its	O
eigenspaces	O
is	O
equal	O
to	O
,	O
which	O
is	O
the	O
case	O
if	O
and	O
only	O
if	O
there	O
exists	O
a	O
basis	O
of	O
consisting	O
of	O
eigenvectors	O
of	O
.	O
</s>
<s>
If	O
such	O
a	O
basis	O
has	O
been	O
found	O
,	O
one	O
can	O
form	O
the	O
matrix	B-Architecture
having	O
these	O
basis	O
vectors	O
as	O
columns	O
,	O
and	O
will	O
be	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
whose	O
diagonal	O
entries	O
are	O
the	O
eigenvalues	O
of	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
is	O
known	O
as	O
a	O
modal	B-Algorithm
matrix	I-Algorithm
for	O
.	O
</s>
<s>
A	O
linear	B-Architecture
map	I-Architecture
is	O
diagonalizable	B-Algorithm
if	O
and	O
only	O
if	O
the	O
sum	O
of	O
the	O
dimensions	O
of	O
its	O
eigenspaces	O
is	O
equal	O
to	O
which	O
is	O
the	O
case	O
if	O
and	O
only	O
if	O
there	O
exists	O
a	O
basis	O
of	O
consisting	O
of	O
eigenvectors	O
of	O
.	O
</s>
<s>
With	O
respect	O
to	O
such	O
a	O
basis	O
,	O
will	O
be	O
represented	O
by	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
An	O
matrix	B-Architecture
is	O
diagonalizable	B-Algorithm
over	O
the	O
field	O
if	O
it	O
has	O
distinct	O
eigenvalues	O
in	O
i.e.	O
</s>
<s>
Consider	O
which	O
has	O
eigenvalues	O
1	O
,	O
2	O
,	O
2	O
(	O
not	O
all	O
distinct	O
)	O
and	O
is	O
diagonalizable	B-Algorithm
with	O
diagonal	O
form	O
(	O
similar	B-Algorithm
to	O
and	O
change	O
of	O
basis	O
matrix	B-Architecture
:	O
The	O
converse	O
fails	O
when	O
has	O
an	O
eigenspace	O
of	O
dimension	O
higher	O
than	O
1	O
.	O
</s>
<s>
A	O
linear	B-Architecture
map	I-Architecture
with	O
is	O
diagonalizable	B-Algorithm
if	O
it	O
has	O
distinct	O
eigenvalues	O
,	O
i.e.	O
</s>
<s>
Let	O
be	O
a	O
matrix	B-Architecture
over	O
If	O
is	O
diagonalizable	B-Algorithm
,	O
then	O
so	O
is	O
any	O
power	O
of	O
it	O
.	O
</s>
<s>
Conversely	O
,	O
if	O
is	O
invertible	O
,	O
is	O
algebraically	O
closed	O
,	O
and	O
is	O
diagonalizable	B-Algorithm
for	O
some	O
that	O
is	O
not	O
an	O
integer	O
multiple	O
of	O
the	O
characteristic	O
of	O
then	O
is	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
Over	O
the	O
complex	O
numbers	O
,	O
almost	O
every	O
matrix	B-Architecture
is	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
More	O
precisely	O
:	O
the	O
set	O
of	O
complex	O
matrices	O
that	O
are	O
not	O
diagonalizable	B-Algorithm
over	O
considered	O
as	O
a	O
subset	O
of	O
has	O
Lebesgue	O
measure	O
zero	O
.	O
</s>
<s>
One	O
can	O
also	O
say	O
that	O
the	O
diagonalizable	B-Algorithm
matrices	I-Algorithm
form	O
a	O
dense	O
subset	O
with	O
respect	O
to	O
the	O
Zariski	O
topology	O
:	O
the	O
non-diagonalizable	O
matrices	O
lie	O
inside	O
the	O
vanishing	O
set	O
of	O
the	O
discriminant	O
of	O
the	O
characteristic	O
polynomial	O
,	O
which	O
is	O
a	O
hypersurface	O
.	O
</s>
<s>
The	O
Jordan	O
–	O
Chevalley	O
decomposition	O
expresses	O
an	O
operator	O
as	O
the	O
sum	O
of	O
its	O
semisimple	O
(	O
i.e.	O
,	O
diagonalizable	B-Algorithm
)	O
part	O
and	O
its	O
nilpotent	O
part	O
.	O
</s>
<s>
Hence	O
,	O
a	O
matrix	B-Architecture
is	O
diagonalizable	B-Algorithm
if	O
and	O
only	O
if	O
its	O
nilpotent	O
part	O
is	O
zero	O
.	O
</s>
<s>
Put	O
in	O
another	O
way	O
,	O
a	O
matrix	B-Architecture
is	O
diagonalizable	B-Algorithm
if	O
each	O
block	O
in	O
its	O
Jordan	O
form	O
has	O
no	O
nilpotent	O
part	O
;	O
i.e.	O
,	O
each	O
"	O
block	O
"	O
is	O
a	O
one-by-one	O
matrix	B-Architecture
.	O
</s>
<s>
If	O
a	O
matrix	B-Architecture
can	O
be	O
diagonalized	B-Algorithm
,	O
that	O
is	O
,	O
</s>
<s>
The	O
invertibility	O
of	O
also	O
suggests	O
that	O
the	O
eigenvectors	O
are	O
linearly	O
independent	O
and	O
form	O
a	O
basis	O
of	O
This	O
is	O
the	O
necessary	O
and	O
sufficient	O
condition	O
for	O
diagonalizability	B-Algorithm
and	O
the	O
canonical	O
approach	O
of	O
diagonalization	O
.	O
</s>
<s>
When	O
a	O
complex	O
matrix	B-Architecture
is	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
(	O
or	O
more	O
generally	O
a	O
normal	B-Algorithm
matrix	I-Algorithm
)	O
,	O
eigenvectors	O
of	O
can	O
be	O
chosen	O
to	O
form	O
an	O
orthonormal	O
basis	O
of	O
and	O
can	O
be	O
chosen	O
to	O
be	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
in	O
addition	O
,	O
is	O
a	O
real	O
symmetric	B-Algorithm
matrix	I-Algorithm
,	O
then	O
its	O
eigenvectors	O
can	O
be	O
chosen	O
to	O
be	O
an	O
orthonormal	O
basis	O
of	O
and	O
can	O
be	O
chosen	O
to	O
be	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
For	O
most	O
practical	O
work	O
matrices	O
are	O
diagonalized	B-Algorithm
numerically	O
using	O
computer	O
software	O
.	O
</s>
<s>
A	O
set	O
of	O
matrices	O
is	O
said	O
to	O
be	O
simultaneously	O
diagonalizable	B-Algorithm
if	O
there	O
exists	O
a	O
single	O
invertible	O
matrix	B-Architecture
such	O
that	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
for	O
every	O
in	O
the	O
set	O
.	O
</s>
<s>
The	O
following	O
theorem	O
characterizes	O
simultaneously	O
diagonalizable	B-Algorithm
matrices	I-Algorithm
:	O
A	O
set	O
of	O
diagonalizable	B-Algorithm
matrices	I-Algorithm
commutes	O
if	O
and	O
only	O
if	O
the	O
set	O
is	O
simultaneously	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
The	O
set	O
of	O
all	O
diagonalizable	B-Algorithm
matrices	I-Algorithm
(	O
over	O
with	O
is	O
not	O
simultaneously	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
are	O
diagonalizable	B-Algorithm
but	O
not	O
simultaneously	O
diagonalizable	B-Algorithm
because	O
they	O
do	O
not	O
commute	O
.	O
</s>
<s>
A	O
set	O
consists	O
of	O
commuting	O
normal	B-Algorithm
matrices	I-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
simultaneously	O
diagonalizable	B-Algorithm
by	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
;	O
that	O
is	O
,	O
there	O
exists	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
such	O
that	O
is	O
diagonal	O
for	O
every	O
in	O
the	O
set	O
.	O
</s>
<s>
In	O
the	O
language	O
of	O
Lie	O
theory	O
,	O
a	O
set	O
of	O
simultaneously	O
diagonalizable	B-Algorithm
matrices	I-Algorithm
generates	O
a	O
toral	O
Lie	O
algebra	O
.	O
</s>
<s>
Involutions	B-Algorithm
are	O
diagonalizable	B-Algorithm
over	O
the	O
reals	O
(	O
and	O
indeed	O
any	O
field	O
of	O
characteristic	O
not	O
2	O
)	O
,	O
with	O
±1	O
on	O
the	O
diagonal	O
.	O
</s>
<s>
Finite	O
order	O
endomorphisms	O
are	O
diagonalizable	B-Algorithm
over	O
(	O
or	O
any	O
algebraically	O
closed	O
field	O
where	O
the	O
characteristic	O
of	O
the	O
field	O
does	O
not	O
divide	O
the	O
order	O
of	O
the	O
endomorphism	O
)	O
with	O
roots	O
of	O
unity	O
on	O
the	O
diagonal	O
.	O
</s>
<s>
Projections	B-Algorithm
are	O
diagonalizable	B-Algorithm
,	O
with	O
0s	O
and	O
1s	O
on	O
the	O
diagonal	O
.	O
</s>
<s>
Real	O
symmetric	B-Algorithm
matrices	I-Algorithm
are	O
diagonalizable	B-Algorithm
by	O
orthogonal	B-Algorithm
matrices	I-Algorithm
;	O
i.e.	O
,	O
given	O
a	O
real	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
diagonal	O
for	O
some	O
orthogonal	B-Algorithm
matrix	I-Algorithm
More	O
generally	O
,	O
matrices	O
are	O
diagonalizable	B-Algorithm
by	O
unitary	B-Algorithm
matrices	I-Algorithm
if	O
and	O
only	O
if	O
they	O
are	O
normal	B-Algorithm
.	O
</s>
<s>
In	O
the	O
case	O
of	O
the	O
real	O
symmetric	B-Algorithm
matrix	I-Algorithm
,	O
we	O
see	O
that	O
so	O
clearly	O
holds	O
.	O
</s>
<s>
Examples	O
of	O
normal	B-Algorithm
matrices	I-Algorithm
are	O
real	O
symmetric	B-Algorithm
(	O
or	O
skew-symmetric	B-Algorithm
)	O
matrices	O
(	O
e.g.	O
</s>
<s>
covariance	O
matrices	O
)	O
and	O
Hermitian	B-Algorithm
matrices	I-Algorithm
(	O
or	O
skew-Hermitian	O
matrices	O
)	O
.	O
</s>
<s>
In	O
general	O
,	O
a	O
rotation	B-Algorithm
matrix	I-Algorithm
is	O
not	O
diagonalizable	B-Algorithm
over	O
the	O
reals	O
,	O
but	O
all	O
rotation	B-Algorithm
matrices	I-Algorithm
are	O
diagonalizable	B-Algorithm
over	O
the	O
complex	O
field	O
.	O
</s>
<s>
Even	O
if	O
a	O
matrix	B-Architecture
is	O
not	O
diagonalizable	B-Algorithm
,	O
it	O
is	O
always	O
possible	O
to	O
"	O
do	O
the	O
best	O
one	O
can	O
"	O
,	O
and	O
find	O
a	O
matrix	B-Architecture
with	O
the	O
same	O
properties	O
consisting	O
of	O
eigenvalues	O
on	O
the	O
leading	O
diagonal	O
,	O
and	O
either	O
ones	O
or	O
zeroes	O
on	O
the	O
superdiagonal	O
–	O
known	O
as	O
Jordan	O
normal	B-Algorithm
form	O
.	O
</s>
<s>
Some	O
matrices	O
are	O
not	O
diagonalizable	B-Algorithm
over	O
any	O
field	O
,	O
most	O
notably	O
nonzero	O
nilpotent	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
This	O
matrix	B-Architecture
is	O
not	O
diagonalizable	B-Algorithm
:	O
there	O
is	O
no	O
matrix	B-Architecture
such	O
that	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Some	O
real	B-Architecture
matrices	I-Architecture
are	O
not	O
diagonalizable	B-Algorithm
over	O
the	O
reals	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
does	O
not	O
have	O
any	O
real	O
eigenvalues	O
,	O
so	O
there	O
is	O
no	O
real	O
matrix	B-Architecture
such	O
that	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
the	O
above	O
examples	O
show	O
that	O
the	O
sum	O
of	O
diagonalizable	B-Algorithm
matrices	I-Algorithm
need	O
not	O
be	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
Diagonalizing	B-Algorithm
a	O
matrix	B-Architecture
is	O
the	O
same	O
process	O
as	O
finding	O
its	O
eigenvalues	O
and	O
eigenvectors	O
,	O
in	O
the	O
case	O
that	O
the	O
eigenvectors	O
form	O
a	O
basis	O
.	O
</s>
<s>
The	O
roots	O
of	O
the	O
characteristic	O
polynomial	O
are	O
the	O
eigenvalues	O
Solving	O
the	O
linear	O
system	O
gives	O
the	O
eigenvectors	O
and	O
while	O
gives	O
that	O
is	O
,	O
for	O
These	O
vectors	O
form	O
a	O
basis	O
of	O
so	O
we	O
can	O
assemble	O
them	O
as	O
the	O
column	O
vectors	O
of	O
a	O
change-of-basis	O
matrix	B-Architecture
to	O
get	O
:	O
</s>
<s>
and	O
the	O
latter	O
is	O
easy	O
to	O
calculate	O
since	O
it	O
only	O
involves	O
the	O
powers	O
of	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
for	O
the	O
matrix	B-Architecture
with	O
eigenvalues	O
in	O
the	O
example	O
above	O
we	O
compute	O
:	O
</s>
<s>
This	O
approach	O
can	O
be	O
generalized	O
to	O
matrix	B-Architecture
exponential	O
and	O
other	O
matrix	B-Architecture
functions	O
that	O
can	O
be	O
defined	O
as	O
power	O
series	O
.	O
</s>
<s>
For	O
example	O
,	O
consider	O
the	O
following	O
matrix	B-Architecture
:	O
</s>
<s>
In	O
quantum	O
mechanical	O
and	O
quantum	O
chemical	O
computations	O
matrix	B-Architecture
diagonalization	O
is	O
one	O
of	O
the	O
most	O
frequently	O
applied	O
numerical	O
processes	O
.	O
</s>
<s>
A	O
very	O
common	O
approximation	O
is	O
to	O
truncate	O
Hilbert	O
space	O
to	O
finite	O
dimension	O
,	O
after	O
which	O
the	O
Schrödinger	O
equation	O
can	O
be	O
formulated	O
as	O
an	O
eigenvalue	O
problem	O
of	O
a	O
real	O
symmetric	B-Algorithm
,	O
or	O
complex	O
Hermitian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
First-order	O
perturbation	O
theory	O
also	O
leads	O
to	O
matrix	B-Architecture
eigenvalue	O
problem	O
for	O
degenerate	O
states	O
.	O
</s>
