<s>
Bohemian	B-Algorithm
matrices	I-Algorithm
can	O
have	O
other	O
structures	O
as	O
well	O
,	O
such	O
as	O
being	O
a	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
or	O
an	O
upper	B-Algorithm
Hessenberg	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
One	O
common	O
application	O
is	O
for	O
testing	O
software	O
for	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
Bohemian	B-Algorithm
matrices	I-Algorithm
are	O
(	O
usually	O
)	O
exactly	O
representable	O
on	O
a	O
computer	O
,	O
and	O
one	O
can	O
search	O
for	O
extreme	O
behavior	O
either	O
by	O
exhaustive	O
search	O
(	O
for	O
small	O
dimension	O
)	O
,	O
by	O
random	O
sampling	O
,	O
or	O
by	O
optimization	O
techniques	O
.	O
</s>
<s>
a	O
number	O
of	O
optimal	O
(	O
or	O
just	O
interesting	O
)	O
classes	O
of	O
Bohemian	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
In	O
a	O
talk	O
given	O
at	O
the	O
2018	O
Bohemian	B-Algorithm
Matrices	I-Algorithm
and	O
Applications	O
Workshop	O
,	O
Nick	O
Higham	O
explained	O
how	O
he	O
had	O
used	O
genetic	B-Algorithm
algorithms	I-Algorithm
on	O
Bohemian	B-Algorithm
matrices	I-Algorithm
with	O
population	O
to	O
sharpen	O
lower	O
bounds	O
on	O
the	O
maximal	O
growth	O
factor	O
for	O
rook	O
pivoting	O
.	O
</s>
<s>
Bohemian	B-Algorithm
matrices	I-Algorithm
can	O
be	O
studied	O
by	O
random	O
sampling	O
,	O
and	O
such	O
studies	O
might	O
be	O
considered	O
as	O
part	O
of	O
the	O
field	O
of	O
Random	O
Matrices	B-Architecture
.	O
</s>
<s>
However	O
,	O
most	O
of	O
the	O
focus	O
of	O
the	O
study	O
of	O
Random	O
Matrices	B-Architecture
to	O
date	O
has	O
been	O
on	O
real	O
symmetric	O
or	O
Hermitian	B-Algorithm
matrices	I-Algorithm
,	O
or	O
on	O
the	O
study	O
of	O
matrices	B-Architecture
whose	O
entries	O
are	O
drawn	O
from	O
a	O
continuous	O
distribution	O
such	O
as	O
Gaussian	O
ensembles	O
.	O
</s>
<s>
Matrices	B-Architecture
with	O
entries	O
only	O
from	O
are	O
sometimes	O
called	O
Bernoulli	O
matrices	B-Architecture
,	O
which	O
are	O
therefore	O
examples	O
of	O
Bohemian	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Therefore	O
Hadamard	O
matrices	B-Architecture
are	O
also	O
Bohemian	O
.	O
</s>
<s>
Hadamard	O
matrices	B-Architecture
(	O
and	O
indeed	O
Bernoulli	O
matrices	B-Architecture
)	O
have	O
been	O
studied	O
for	O
much	O
longer	O
than	O
the	O
name	O
"	O
Bohemian	O
matrix	O
"	O
has	O
existed	O
,	O
of	O
course	O
.	O
</s>
<s>
But	O
the	O
kinds	O
of	O
questions	O
one	O
can	O
ask	O
about	O
Hadamard	O
matrices	B-Architecture
(	O
the	O
original	O
question	O
was	O
which	O
matrices	B-Architecture
have	O
maximal	O
determinant	O
)	O
can	O
also	O
be	O
asked	O
about	O
other	O
Bohemian	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
One	O
of	O
the	O
first	O
generalizations	O
of	O
Hadamard	O
matrices	B-Architecture
was	O
to	O
what	O
are	O
now	O
sometimes	O
called	O
Butson-type	B-Algorithm
Hadamard	I-Algorithm
matrices	I-Algorithm
,	O
whose	O
entries	O
are	O
th	O
roots	O
of	O
unity	O
for	O
.	O
</s>
<s>
These	O
can	O
also	O
be	O
considered	O
prototypical	O
Bohemian	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Matrices	B-Architecture
with	O
discrete	O
entries	O
,	O
especially	O
incidence	B-Algorithm
matrices	I-Algorithm
,	O
are	O
crucial	O
to	O
the	O
understanding	O
of	O
Graph	O
Theory	O
.	O
</s>
<s>
A	O
significant	O
number	O
of	O
the	O
open	O
problems	O
listed	O
at	O
the	O
under	O
Bohemian	B-Algorithm
matrices	I-Algorithm
are	O
combinatorial	O
in	O
nature	O
.	O
</s>
<s>
For	O
instance	O
,	O
lists	O
a	O
table	O
of	O
the	O
number	O
of	O
distinct	O
minimal	O
polynomials	O
for	O
Bernoulli	O
matrices	B-Architecture
(	O
that	O
is	O
,	O
Bohemian	B-Algorithm
matrices	I-Algorithm
with	O
population	O
)	O
up	O
to	O
and	O
including	O
dimension	O
5	O
.	O
</s>
<s>
The	O
number	O
of	O
Bernoulli	O
matrices	B-Architecture
of	O
dimension	O
6	O
is	O
and	O
while	O
this	O
set	O
likely	O
could	O
be	O
searched	O
exhaustively	O
(	O
it	O
is	O
delightfully	B-Operating_System
parallel	I-Operating_System
)	O
,	O
the	O
greater-than-exponential	O
growth	O
of	O
the	O
number	O
of	O
matrices	B-Architecture
soon	O
defeats	O
brute	O
force	O
.	O
</s>
<s>
There	O
are	O
symmetries	O
that	O
might	O
be	O
taken	O
advantage	O
of	O
,	O
as	O
is	O
done	O
in	O
for	O
zero-one	O
matrices	B-Architecture
,	O
but	O
these	O
seem	O
to	O
require	O
sophisticated	O
combinatorial	O
knowledge	O
.	O
</s>
<s>
By	O
using	O
companion	B-Algorithm
matrices	I-Algorithm
each	O
of	O
these	O
restricted-coefficient	O
polynomial	O
problems	O
can	O
be	O
considered	O
to	O
be	O
a	O
Bohemian	O
matrix	O
problem	O
.	O
</s>
<s>
The	O
following	O
paper	O
explicitly	O
connects	O
Bohemian	B-Algorithm
matrices	I-Algorithm
to	O
quadratic	O
forms	O
.	O
</s>
<s>
A	O
commonly-used	O
technique	O
to	O
find	O
polynomial	O
roots	O
is	O
to	O
construct	O
a	O
companion	B-Algorithm
matrix	I-Algorithm
for	O
it	O
and	O
use	O
numerical	O
eigenvalue	O
solvers	O
to	O
find	O
the	O
eigenvalues	O
,	O
which	O
are	O
then	O
the	O
roots	O
of	O
the	O
original	O
polynomial	O
.	O
</s>
<s>
Sometimes	O
this	O
does	O
not	O
work	O
well	O
,	O
however	O
,	O
if	O
the	O
polynomial	O
coefficients	O
are	O
at	O
all	O
large	O
,	O
or	O
when	O
the	O
polynomial	O
is	O
otherwise	O
ill-conditioned	B-Algorithm
.	O
</s>
<s>
One	O
can	O
sometimes	O
improve	O
the	O
situation	O
by	O
finding	O
a	O
minimal	O
height	O
companion	B-Algorithm
matrix	I-Algorithm
for	O
the	O
polynomial	O
by	O
looking	O
for	O
such	O
in	O
a	O
Bohemian	O
matrix	O
family	O
.	O
</s>
<s>
entries	O
and	O
are	O
thus	O
special	O
matrices	B-Architecture
.	O
</s>
<s>
The	O
idea	O
is	O
somewhat	O
similar	O
to	O
that	O
of	O
sign	O
pattern	O
matrices	B-Architecture
in	O
which	O
two	O
matrices	B-Architecture
with	O
real	O
entries	O
are	O
considered	O
equivalent	O
if	O
the	O
corresponding	O
entries	O
have	O
the	O
same	O
sign	O
,	O
and	O
that	O
theory	O
looks	O
for	O
common	O
properties	O
.	O
</s>
<s>
A	O
Bohemian	O
matrix	O
with	O
population	O
is	O
an	O
example	O
of	O
a	O
sign	O
pattern	O
matrix	O
and	O
so	O
it	O
has	O
all	O
the	O
properties	O
of	O
such	O
matrices	B-Architecture
;	O
but	O
it	O
may	O
also	O
have	O
special	O
properties	O
owing	O
to	O
its	O
Bohemian	O
nature	O
.	O
</s>
