<s>
In	O
linear	B-Language
algebra	I-Language
,	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
is	O
a	O
band	B-Algorithm
matrix	I-Algorithm
that	O
has	O
nonzero	O
elements	O
only	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
,	O
the	O
subdiagonal/lower	O
diagonal	O
(	O
the	O
first	O
diagonal	O
below	O
this	O
)	O
,	O
and	O
the	O
supradiagonal/upper	O
diagonal	O
(	O
the	O
first	O
diagonal	O
above	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
)	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
following	O
matrix	B-Architecture
is	O
tridiagonal	B-Language
:	O
</s>
<s>
The	O
determinant	O
of	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
is	O
given	O
by	O
the	O
continuant	O
of	O
its	O
elements	O
.	O
</s>
<s>
An	O
orthogonal	O
transformation	O
of	O
a	O
symmetric	B-Algorithm
(	O
or	O
Hermitian	B-Algorithm
)	O
matrix	B-Architecture
to	O
tridiagonal	B-Language
form	O
can	O
be	O
done	O
with	O
the	O
Lanczos	O
algorithm	O
.	O
</s>
<s>
A	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
that	O
is	O
both	O
upper	O
and	O
lower	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
is	O
a	O
direct	O
sum	O
of	O
p	O
1-by-1	O
and	O
q	O
2-by-2	O
matrices	O
such	O
that	O
—	O
the	O
dimension	O
of	O
the	O
tridiagonal	B-Language
.	O
</s>
<s>
Although	O
a	O
general	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
is	O
not	O
necessarily	O
symmetric	B-Algorithm
or	O
Hermitian	B-Algorithm
,	O
many	O
of	O
those	O
that	O
arise	O
when	O
solving	O
linear	B-Language
algebra	I-Language
problems	O
have	O
one	O
of	O
these	O
properties	O
.	O
</s>
<s>
Furthermore	O
,	O
if	O
a	O
real	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
A	O
satisfies	O
ak	O
,	O
k+1	O
ak+1	O
,	O
k	O
>	O
0	O
for	O
all	O
k	O
,	O
so	O
that	O
the	O
signs	O
of	O
its	O
entries	O
are	O
symmetric	B-Algorithm
,	O
then	O
it	O
is	O
similar	B-Algorithm
to	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
,	O
by	O
a	O
diagonal	O
change	O
of	O
basis	O
matrix	B-Architecture
.	O
</s>
<s>
If	O
we	O
replace	O
the	O
strict	O
inequality	O
by	O
ak	O
,	O
k+1	O
ak+1	O
,	O
k	O
0	O
,	O
then	O
by	O
continuity	O
,	O
the	O
eigenvalues	O
are	O
still	O
guaranteed	O
to	O
be	O
real	O
,	O
but	O
the	O
matrix	B-Architecture
need	O
no	O
longer	O
be	O
similar	B-Algorithm
to	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Many	O
linear	B-Language
algebra	I-Language
algorithms	O
require	O
significantly	O
less	O
computational	O
effort	O
when	O
applied	O
to	O
diagonal	O
matrices	O
,	O
and	O
this	O
improvement	O
often	O
carries	O
over	O
to	O
tridiagonal	B-Language
matrices	O
as	O
well	O
.	O
</s>
<s>
The	O
determinant	O
of	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
A	O
of	O
order	O
n	O
can	O
be	O
computed	O
from	O
a	O
three-term	O
recurrence	O
relation	O
.	O
</s>
<s>
The	O
cost	O
of	O
computing	O
the	O
determinant	O
of	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
using	O
this	O
formula	O
is	O
linear	O
in	O
n	O
,	O
while	O
the	O
cost	O
is	O
cubic	O
for	O
a	O
general	O
matrix	B-Architecture
.	O
</s>
<s>
Closed	O
form	O
solutions	O
can	O
be	O
computed	O
for	O
special	O
cases	O
such	O
as	O
symmetric	B-Algorithm
matrices	I-Algorithm
with	O
all	O
diagonal	O
and	O
off-diagonal	O
elements	O
equal	O
or	O
Toeplitz	B-Algorithm
matrices	I-Algorithm
and	O
for	O
the	O
general	O
case	O
as	O
well	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
inverse	O
of	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
is	O
a	O
semiseparable	O
matrix	B-Architecture
and	O
vice	O
versa	O
.	O
</s>
<s>
A	O
system	O
of	O
equations	O
Ax	O
=	O
b	O
for	O
can	O
be	O
solved	O
by	O
an	O
efficient	O
form	O
of	O
Gaussian	O
elimination	O
when	O
A	O
is	O
tridiagonal	B-Language
called	O
tridiagonal	B-Language
matrix	I-Language
algorithm	I-Language
,	O
requiring	O
O(n )	O
operations	O
.	O
</s>
<s>
When	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
is	O
also	O
Toeplitz	B-Algorithm
,	O
there	O
is	O
a	O
simple	O
closed-form	O
solution	O
for	O
its	O
eigenvalues	O
,	O
namely	O
:	O
</s>
<s>
A	O
real	O
symmetric	B-Algorithm
tridiagonal	B-Algorithm
matrix	I-Algorithm
has	O
real	O
eigenvalues	O
,	O
and	O
all	O
the	O
eigenvalues	O
are	O
distinct	O
(	O
simple	O
)	O
if	O
all	O
off-diagonal	O
elements	O
are	O
nonzero	O
.	O
</s>
<s>
Numerous	O
methods	O
exist	O
for	O
the	O
numerical	O
computation	O
of	O
the	O
eigenvalues	O
of	O
a	O
real	O
symmetric	B-Algorithm
tridiagonal	B-Algorithm
matrix	I-Algorithm
to	O
arbitrary	O
finite	O
precision	O
,	O
typically	O
requiring	O
operations	O
for	O
a	O
matrix	B-Architecture
of	O
size	O
,	O
although	O
fast	O
algorithms	O
exist	O
which	O
(	O
without	O
parallel	O
computation	O
)	O
require	O
only	O
.	O
</s>
<s>
As	O
a	O
side	O
note	O
,	O
an	O
unreduced	O
symmetric	B-Algorithm
tridiagonal	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
containing	O
non-zero	O
off-diagonal	O
elements	O
of	O
the	O
tridiagonal	B-Language
,	O
where	O
the	O
eigenvalues	O
are	O
distinct	O
while	O
the	O
eigenvectors	O
are	O
unique	O
up	O
to	O
a	O
scale	O
factor	O
and	O
are	O
mutually	O
orthogonal	O
.	O
</s>
<s>
For	O
unsymmetric	O
or	O
nonsymmetric	O
tridiagonal	B-Language
matrices	O
one	O
can	O
compute	O
the	O
eigendecomposition	O
using	O
a	O
similarity	O
transformation	O
.	O
</s>
<s>
The	O
similarity	O
transformation	O
yields	O
a	O
symmetric	B-Algorithm
tridiagonal	B-Algorithm
matrix	I-Algorithm
by	O
:	O
</s>
<s>
A	O
transformation	O
that	O
reduces	O
a	O
general	O
matrix	B-Architecture
to	O
Hessenberg	B-Algorithm
form	I-Algorithm
will	O
reduce	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
to	O
tridiagonal	B-Language
form	O
.	O
</s>
<s>
So	O
,	O
many	O
eigenvalue	O
algorithms	O
,	O
when	O
applied	O
to	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
,	O
reduce	O
the	O
input	O
Hermitian	B-Algorithm
matrix	I-Algorithm
to	O
(	O
symmetric	B-Algorithm
real	O
)	O
tridiagonal	B-Language
form	O
as	O
a	O
first	O
step	O
.	O
</s>
<s>
A	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
can	O
also	O
be	O
stored	O
more	O
efficiently	O
than	O
a	O
general	O
matrix	B-Architecture
by	O
using	O
a	O
special	O
storage	B-Algorithm
scheme	I-Algorithm
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
LAPACK	B-Application
Fortran	B-Application
package	O
stores	O
an	O
unsymmetric	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
of	O
order	O
n	O
in	O
three	O
one-dimensional	O
arrays	O
,	O
one	O
of	O
length	O
n	O
containing	O
the	O
diagonal	O
elements	O
,	O
and	O
two	O
of	O
length	O
n	O
1	O
containing	O
the	O
subdiagonal	O
and	O
superdiagonal	O
elements	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
is	O
tridiagonal	B-Language
with	O
and	O
.	O
</s>
