<s>
In	O
mathematics	O
,	O
an	O
eigenfunction	B-Algorithm
of	O
a	O
linear	B-Architecture
operator	I-Architecture
D	O
defined	O
on	O
some	O
function	B-Algorithm
space	I-Algorithm
is	O
any	O
non-zero	O
function	O
in	O
that	O
space	O
that	O
,	O
when	O
acted	O
upon	O
by	O
D	O
,	O
is	O
only	O
multiplied	O
by	O
some	O
scaling	O
factor	O
called	O
an	O
eigenvalue	O
.	O
</s>
<s>
for	O
some	O
scalar	O
eigenvalue	O
The	O
solutions	O
to	O
this	O
equation	O
may	O
also	O
be	O
subject	O
to	O
boundary	O
conditions	O
that	O
limit	O
the	O
allowable	O
eigenvalues	O
and	O
eigenfunctions	B-Algorithm
.	O
</s>
<s>
An	O
eigenfunction	B-Algorithm
is	O
a	O
type	O
of	O
eigenvector	O
.	O
</s>
<s>
In	O
general	O
,	O
an	O
eigenvector	O
of	O
a	O
linear	B-Architecture
operator	I-Architecture
D	O
defined	O
on	O
some	O
vector	O
space	O
is	O
a	O
nonzero	O
vector	O
in	O
the	O
domain	O
of	O
D	O
that	O
,	O
when	O
D	O
acts	O
upon	O
it	O
,	O
is	O
simply	O
scaled	O
by	O
some	O
scalar	O
value	O
called	O
an	O
eigenvalue	O
.	O
</s>
<s>
In	O
the	O
special	O
case	O
where	O
D	O
is	O
defined	O
on	O
a	O
function	B-Algorithm
space	I-Algorithm
,	O
the	O
eigenvectors	O
are	O
referred	O
to	O
as	O
eigenfunctions	B-Algorithm
.	O
</s>
<s>
Each	O
value	O
of	O
λ	O
corresponds	O
to	O
one	O
or	O
more	O
eigenfunctions	B-Algorithm
.	O
</s>
<s>
If	O
multiple	O
linearly	O
independent	O
eigenfunctions	B-Algorithm
have	O
the	O
same	O
eigenvalue	O
,	O
the	O
eigenvalue	O
is	O
said	O
to	O
be	O
degenerate	O
and	O
the	O
maximum	O
number	O
of	O
linearly	O
independent	O
eigenfunctions	B-Algorithm
associated	O
with	O
the	O
same	O
eigenvalue	O
is	O
the	O
eigenvalue	O
's	O
degree	O
of	O
degeneracy	O
or	O
geometric	O
multiplicity	O
.	O
</s>
<s>
is	O
the	O
eigenfunction	B-Algorithm
of	O
the	O
derivative	O
operator	O
,	O
where	O
f0	O
is	O
a	O
parameter	O
that	O
depends	O
on	O
the	O
boundary	O
conditions	O
.	O
</s>
<s>
Note	O
that	O
in	O
this	O
case	O
the	O
eigenfunction	B-Algorithm
is	O
itself	O
a	O
function	O
of	O
its	O
associated	O
eigenvalue	O
λ	O
,	O
which	O
can	O
take	O
any	O
real	O
or	O
complex	O
value	O
.	O
</s>
<s>
In	O
particular	O
,	O
note	O
that	O
for	O
λ	O
=	O
0	O
the	O
eigenfunction	B-Algorithm
f(t )	O
is	O
a	O
constant	O
.	O
</s>
<s>
Eigenfunctions	B-Algorithm
can	O
be	O
expressed	O
as	O
column	O
vectors	O
and	O
linear	B-Architecture
operators	I-Architecture
can	O
be	O
expressed	O
as	O
matrices	O
,	O
although	O
they	O
may	O
have	O
infinite	O
dimensions	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
many	O
of	O
the	O
concepts	O
related	O
to	O
eigenvectors	O
of	O
matrices	O
carry	O
over	O
to	O
the	O
study	O
of	O
eigenfunctions	B-Algorithm
.	O
</s>
<s>
Suppose	O
the	O
function	B-Algorithm
space	I-Algorithm
has	O
an	O
orthonormal	O
basis	O
given	O
by	O
the	O
set	O
of	O
functions	O
{	O
u1(t )	O
,	O
u2(t )	O
,	O
…,	O
un(t )	O
}	O
,	O
where	O
n	O
may	O
be	O
infinite	O
.	O
</s>
<s>
where	O
δij	O
is	O
the	O
Kronecker	O
delta	O
and	O
can	O
be	O
thought	O
of	O
as	O
the	O
elements	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
f(t )	O
is	O
an	O
eigenfunction	B-Algorithm
of	O
D	O
with	O
eigenvalue	O
λ	O
,	O
then	O
Ab	O
=	O
λb	O
.	O
</s>
<s>
Suppose	O
the	O
linear	B-Architecture
operator	I-Architecture
D	O
acts	O
on	O
a	O
function	B-Algorithm
space	I-Algorithm
that	O
is	O
a	O
Hilbert	O
space	O
with	O
an	O
orthonormal	O
basis	O
given	O
by	O
the	O
set	O
of	O
functions	O
{	O
u1(t )	O
,	O
u2(t )	O
,	O
…,	O
un(t )	O
}	O
,	O
where	O
n	O
may	O
be	O
infinite	O
.	O
</s>
<s>
By	O
analogy	O
with	O
Hermitian	B-Algorithm
matrices	I-Algorithm
,	O
D	O
is	O
a	O
Hermitian	O
operator	O
if	O
Aij	O
=	O
Aji*	O
,	O
or	O
:	O
</s>
<s>
Consider	O
the	O
Hermitian	O
operator	O
D	O
with	O
eigenvalues	O
λ1	O
,	O
λ2	O
,	O
…	O
and	O
corresponding	O
eigenfunctions	B-Algorithm
f1(t )	O
,	O
f2(t )	O
,	O
…	O
.	O
</s>
<s>
For	O
degenerate	O
eigenfunctions	B-Algorithm
with	O
the	O
same	O
eigenvalue	O
λi	O
,	O
orthogonal	O
eigenfunctions	B-Algorithm
can	O
always	O
be	O
chosen	O
that	O
span	O
the	O
eigenspace	O
associated	O
with	O
λi	O
,	O
for	O
example	O
by	O
using	O
the	O
Gram-Schmidt	B-Algorithm
process	I-Algorithm
.	O
</s>
<s>
Depending	O
on	O
whether	O
the	O
spectrum	O
is	O
discrete	O
or	O
continuous	O
,	O
the	O
eigenfunctions	B-Algorithm
can	O
be	O
normalized	O
by	O
setting	O
the	O
inner	O
product	O
of	O
the	O
eigenfunctions	B-Algorithm
equal	O
to	O
either	O
a	O
Kronecker	O
delta	O
or	O
a	O
Dirac	O
delta	O
function	O
,	O
respectively	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
in	O
many	O
important	O
cases	O
,	O
the	O
eigenfunctions	B-Algorithm
of	O
the	O
Hermitian	O
operator	O
form	O
an	O
orthonormal	O
basis	O
.	O
</s>
<s>
In	O
these	O
cases	O
,	O
an	O
arbitrary	O
function	O
can	O
be	O
expressed	O
as	O
a	O
linear	O
combination	O
of	O
the	O
eigenfunctions	B-Algorithm
of	O
the	O
Hermitian	O
operator	O
.	O
</s>
<s>
The	O
eigenfunctions	B-Algorithm
of	O
the	O
Hamiltonian	O
operator	O
are	O
stationary	O
states	O
of	O
the	O
quantum	O
mechanical	O
system	O
,	O
each	O
with	O
a	O
corresponding	O
energy	O
.	O
</s>
<s>
The	O
Hamiltonian	O
operator	O
is	O
an	O
example	O
of	O
a	O
Hermitian	O
operator	O
whose	O
eigenfunctions	B-Algorithm
form	O
an	O
orthonormal	O
basis	O
.	O
</s>
<s>
In	O
the	O
study	O
of	O
signals	O
and	O
systems	O
,	O
an	O
eigenfunction	B-Algorithm
of	O
a	O
system	O
is	O
a	O
signal	O
that	O
,	O
when	O
input	O
into	O
the	O
system	O
,	O
produces	O
a	O
response	O
,	O
where	O
is	O
a	O
complex	O
scalar	O
eigenvalue	O
.	O
</s>
