<s>
In	O
linear	B-Language
algebra	I-Language
,	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
or	O
diagonal-constant	O
matrix	B-Architecture
,	O
named	O
after	O
Otto	O
Toeplitz	O
,	O
is	O
a	O
matrix	B-Architecture
in	O
which	O
each	O
descending	O
diagonal	O
from	O
left	O
to	O
right	O
is	O
constant	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
following	O
matrix	B-Architecture
is	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
is	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
is	O
not	O
necessarily	O
square	B-Algorithm
.	O
</s>
<s>
is	O
called	O
a	O
Toeplitz	O
system	O
if	O
is	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Toeplitz	O
systems	O
can	O
be	O
solved	O
by	O
the	O
Levinson	B-Algorithm
algorithm	I-Algorithm
in	O
time	O
.	O
</s>
<s>
they	O
exhibit	O
numerical	B-Algorithm
stability	I-Algorithm
for	O
well-conditioned	B-Algorithm
linear	O
systems	O
)	O
.	O
</s>
<s>
The	O
algorithm	O
can	O
also	O
be	O
used	O
to	O
find	O
the	O
determinant	O
of	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
in	O
time	O
.	O
</s>
<s>
A	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
can	O
also	O
be	O
decomposed	O
(	O
i.e.	O
</s>
<s>
An	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
may	O
be	O
defined	O
as	O
a	O
matrix	B-Architecture
where	O
,	O
for	O
constants	O
.	O
</s>
<s>
The	O
set	O
of	O
Toeplitz	B-Algorithm
matrices	I-Algorithm
is	O
a	O
subspace	O
of	O
the	O
vector	O
space	O
of	O
matrices	O
(	O
under	O
matrix	B-Architecture
addition	O
and	O
scalar	O
multiplication	O
)	O
.	O
</s>
<s>
Two	O
Toeplitz	B-Algorithm
matrices	I-Algorithm
may	O
be	O
added	O
in	O
time	O
(	O
by	O
storing	O
only	O
one	O
value	O
of	O
each	O
diagonal	O
)	O
and	O
multiplied	O
in	O
time	O
.	O
</s>
<s>
Toeplitz	B-Algorithm
matrices	I-Algorithm
are	O
persymmetric	B-Algorithm
.	O
</s>
<s>
Symmetric	O
Toeplitz	B-Algorithm
matrices	I-Algorithm
are	O
both	O
centrosymmetric	B-Algorithm
and	O
bisymmetric	B-Algorithm
.	O
</s>
<s>
Toeplitz	B-Algorithm
matrices	I-Algorithm
are	O
also	O
closely	O
connected	O
with	O
Fourier	O
series	O
,	O
because	O
the	O
multiplication	B-Algorithm
operator	I-Algorithm
by	O
a	O
trigonometric	O
polynomial	O
,	O
compressed	B-Algorithm
to	O
a	O
finite-dimensional	O
space	O
,	O
can	O
be	O
represented	O
by	O
such	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
Similarly	O
,	O
one	O
can	O
represent	O
linear	O
convolution	B-Language
as	O
multiplication	O
by	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Toeplitz	B-Algorithm
matrices	I-Algorithm
commute	O
asymptotically	O
.	O
</s>
<s>
This	O
means	O
they	O
diagonalize	B-Algorithm
in	O
the	O
same	O
basis	O
when	O
the	O
row	O
and	O
column	O
dimension	O
tends	O
to	O
infinity	O
.	O
</s>
<s>
where	O
and	O
are	O
lower	O
triangular	O
Toeplitz	B-Algorithm
matrices	I-Algorithm
and	O
is	O
a	O
strictly	O
lower	O
triangular	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
convolution	B-Language
operation	O
can	O
be	O
constructed	O
as	O
a	O
matrix	B-Architecture
multiplication	O
,	O
where	O
one	O
of	O
the	O
inputs	O
is	O
converted	O
into	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
convolution	B-Language
of	O
and	O
can	O
be	O
formulated	O
as	O
:	O
</s>
<s>
A	O
bi-infinite	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
(	O
i.e.	O
</s>
<s>
entries	O
indexed	O
by	O
)	O
induces	O
a	O
linear	B-Architecture
operator	I-Architecture
on	O
.	O
</s>
<s>
The	O
induced	O
operator	O
is	O
bounded	O
if	O
and	O
only	O
if	O
the	O
coefficients	O
of	O
the	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
are	O
the	O
Fourier	O
coefficients	O
of	O
some	O
essentially	O
bounded	O
function	O
.	O
</s>
<s>
In	O
such	O
cases	O
,	O
is	O
called	O
the	O
symbol	O
of	O
the	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
,	O
and	O
the	O
spectral	O
norm	O
of	O
the	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
coincides	O
with	O
the	O
norm	O
of	O
its	O
symbol	O
.	O
</s>
