<s>
In	O
linear	B-Language
algebra	I-Language
,	O
a	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
is	O
a	O
special	O
kind	O
of	O
square	B-Algorithm
matrix	I-Algorithm
,	O
one	O
that	O
is	O
"	O
almost	O
"	O
triangular	B-Algorithm
.	O
</s>
<s>
To	O
be	O
exact	O
,	O
an	O
upper	B-Algorithm
Hessenberg	I-Algorithm
matrix	I-Algorithm
has	O
zero	O
entries	O
below	O
the	O
first	O
subdiagonal	O
,	O
and	O
a	O
lower	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
has	O
zero	O
entries	O
above	O
the	O
first	O
superdiagonal	O
.	O
</s>
<s>
A	O
square	O
matrix	O
is	O
said	O
to	O
be	O
in	O
upper	O
Hessenberg	B-Algorithm
form	I-Algorithm
or	O
to	O
be	O
an	O
upper	B-Algorithm
Hessenberg	I-Algorithm
matrix	I-Algorithm
if	O
for	O
all	O
with	O
.	O
</s>
<s>
An	O
upper	B-Algorithm
Hessenberg	I-Algorithm
matrix	I-Algorithm
is	O
called	O
unreduced	O
if	O
all	O
subdiagonal	O
entries	O
are	O
nonzero	O
,	O
i.e.	O
</s>
<s>
A	O
square	O
matrix	O
is	O
said	O
to	O
be	O
in	O
lower	O
Hessenberg	B-Algorithm
form	I-Algorithm
or	O
to	O
be	O
a	O
lower	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
if	O
its	O
transpose	O
is	O
an	O
upper	B-Algorithm
Hessenberg	I-Algorithm
matrix	I-Algorithm
or	O
equivalently	O
if	O
for	O
all	O
with	O
.	O
</s>
<s>
A	O
lower	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
is	O
called	O
unreduced	O
if	O
all	O
superdiagonal	O
entries	O
are	O
nonzero	O
,	O
i.e.	O
</s>
<s>
The	O
matrix	O
is	O
an	O
upper	O
unreduced	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
,	O
is	O
a	O
lower	O
unreduced	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
and	O
is	O
a	O
lower	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
but	O
is	O
not	O
unreduced	O
.	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
triangular	B-Algorithm
matrices	I-Algorithm
,	O
and	O
this	O
improvement	O
often	O
carries	O
over	O
to	O
Hessenberg	O
matrices	O
as	O
well	O
.	O
</s>
<s>
If	O
the	O
constraints	O
of	O
a	O
linear	B-Language
algebra	I-Language
problem	O
do	O
not	O
allow	O
a	O
general	O
matrix	O
to	O
be	O
conveniently	O
reduced	O
to	O
a	O
triangular	B-Algorithm
one	O
,	O
reduction	O
to	O
Hessenberg	B-Algorithm
form	I-Algorithm
is	O
often	O
the	O
next	O
best	O
thing	O
.	O
</s>
<s>
In	O
fact	O
,	O
reduction	O
of	O
any	O
matrix	O
to	O
a	O
Hessenberg	B-Algorithm
form	I-Algorithm
can	O
be	O
achieved	O
in	O
a	O
finite	O
number	O
of	O
steps	O
(	O
for	O
example	O
,	O
through	O
Householder	B-Algorithm
's	I-Algorithm
transformation	I-Algorithm
of	O
unitary	O
similarity	O
transforms	O
)	O
.	O
</s>
<s>
Subsequent	O
reduction	O
of	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
to	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
can	O
be	O
achieved	O
through	O
iterative	O
procedures	O
,	O
such	O
as	O
shifted	O
QR-factorization	O
.	O
</s>
<s>
In	O
eigenvalue	O
algorithms	O
,	O
the	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
can	O
be	O
further	O
reduced	O
to	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
through	O
Shifted	O
QR-factorization	O
combined	O
with	O
deflation	O
steps	O
.	O
</s>
<s>
Reducing	O
a	O
general	O
matrix	O
to	O
a	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
and	O
then	O
reducing	O
further	O
to	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
,	O
instead	O
of	O
directly	O
reducing	O
a	O
general	O
matrix	O
to	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
,	O
often	O
economizes	O
the	O
arithmetic	O
involved	O
in	O
the	O
QR	O
algorithm	O
for	O
eigenvalue	O
problems	O
.	O
</s>
<s>
Any	O
matrix	O
can	O
be	O
transformed	O
into	O
a	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
by	O
a	O
similarity	O
transformation	O
using	O
Householder	B-Algorithm
transformations	I-Algorithm
.	O
</s>
<s>
The	O
following	O
procedure	O
for	O
such	O
a	O
transformation	O
is	O
adapted	O
from	O
A	O
Second	O
Course	O
In	O
Linear	B-Language
Algebra	I-Language
by	O
Garcia	O
&	O
Roger	O
.	O
</s>
<s>
This	O
householder	B-Algorithm
matrix	I-Algorithm
will	O
map	O
to	O
and	O
as	O
such	O
,	O
the	O
block	O
matrix	O
will	O
map	O
the	O
matrix	O
to	O
the	O
matrix	O
which	O
has	O
only	O
zeros	O
below	O
the	O
second	O
entry	O
of	O
the	O
first	O
column	O
.	O
</s>
<s>
Now	O
construct	O
householder	B-Algorithm
matrix	I-Algorithm
in	O
a	O
similar	O
manner	O
as	O
such	O
that	O
maps	O
the	O
first	O
column	O
of	O
to	O
,	O
where	O
is	O
the	O
submatrix	O
of	O
constructed	O
by	O
removing	O
the	O
first	O
row	O
and	O
the	O
first	O
column	O
of	O
,	O
then	O
let	O
which	O
maps	O
to	O
the	O
matrix	O
which	O
has	O
only	O
zeros	O
below	O
the	O
first	O
and	O
second	O
entry	O
of	O
the	O
subdiagonal	O
.	O
</s>
<s>
Hence	O
,	O
any	O
matrix	O
can	O
be	O
transformed	O
to	O
an	O
upper	B-Algorithm
Hessenberg	I-Algorithm
matrix	I-Algorithm
by	O
a	O
similarity	O
transformation	O
of	O
the	O
form	O
.	O
</s>
<s>
The	O
product	O
of	O
a	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
with	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
is	O
again	O
Hessenberg	O
.	O
</s>
<s>
More	O
precisely	O
,	O
if	O
is	O
upper	O
Hessenberg	O
and	O
is	O
upper	B-Algorithm
triangular	I-Algorithm
,	O
then	O
and	O
are	O
upper	O
Hessenberg	O
.	O
</s>
<s>
A	O
matrix	O
that	O
is	O
both	O
upper	O
Hessenberg	O
and	O
lower	O
Hessenberg	O
is	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
,	O
of	O
which	O
symmetric	O
or	O
Hermitian	O
Hessenberg	O
matrices	O
are	O
important	O
examples	O
.	O
</s>
<s>
The	O
Hessenberg	O
operator	O
is	O
an	O
infinite	O
dimensional	O
Hessenberg	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
It	O
commonly	O
occurs	O
as	O
the	O
generalization	O
of	O
the	O
Jacobi	O
operator	O
to	O
a	O
system	O
of	O
orthogonal	O
polynomials	O
for	O
the	O
space	O
of	O
square-integrable	B-Algorithm
holomorphic	O
functions	O
over	O
some	O
domain	O
—	O
that	O
is	O
,	O
a	O
Bergman	O
space	O
.	O
</s>
