<s>
matrix	B-Architecture
satisfying	O
two	O
conditions	O
:	O
</s>
<s>
orthogonality	B-Algorithm
:	O
,	O
</s>
<s>
where	O
denotes	O
the	O
Hermitian	B-Algorithm
transpose	I-Algorithm
of	O
and	O
is	O
the	O
identity	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
concept	O
is	O
a	O
generalization	O
of	O
the	O
Hadamard	O
matrix	B-Architecture
.	O
</s>
<s>
Note	O
that	O
any	O
complex	B-Algorithm
Hadamard	I-Algorithm
matrix	I-Algorithm
can	O
be	O
made	O
into	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
by	O
multiplying	O
it	O
by	O
;	O
conversely	O
,	O
any	O
unitary	B-Algorithm
matrix	I-Algorithm
whose	O
entries	O
all	O
have	O
modulus	O
becomes	O
a	O
complex	O
Hadamard	O
upon	O
multiplication	O
by	O
.	O
</s>
<s>
Complex	B-Algorithm
Hadamard	I-Algorithm
matrices	I-Algorithm
arise	O
in	O
the	O
study	O
of	O
operator	B-Algorithm
algebras	I-Algorithm
and	O
the	O
theory	O
of	O
quantum	B-Architecture
computation	I-Architecture
.	O
</s>
<s>
Real	O
Hadamard	O
matrices	O
and	O
Butson-type	B-Algorithm
Hadamard	I-Algorithm
matrices	I-Algorithm
form	O
particular	O
cases	O
of	O
complex	B-Algorithm
Hadamard	I-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Complex	B-Algorithm
Hadamard	I-Algorithm
matrices	I-Algorithm
exist	O
for	O
any	O
natural	O
(	O
compare	O
the	O
real	O
case	O
,	O
in	O
which	O
existence	O
is	O
not	O
known	O
for	O
every	O
)	O
.	O
</s>
<s>
Any	O
complex	B-Algorithm
Hadamard	I-Algorithm
matrix	I-Algorithm
is	O
equivalent	O
to	O
a	O
dephased	O
Hadamard	O
matrix	B-Architecture
,	O
in	O
which	O
all	O
elements	O
in	O
the	O
first	O
row	O
and	O
first	O
column	O
are	O
equal	O
to	O
unity	O
.	O
</s>
<s>
For	O
and	O
all	O
complex	B-Algorithm
Hadamard	I-Algorithm
matrices	I-Algorithm
are	O
equivalent	O
to	O
the	O
Fourier	O
matrix	B-Architecture
.	O
</s>
<s>
a	O
continuous	O
,	O
one-parameter	O
family	O
of	O
inequivalent	O
complex	B-Algorithm
Hadamard	I-Algorithm
matrices	I-Algorithm
,	O
</s>
<s>
a	O
one-parameter	O
orbit	O
,	O
including	O
the	O
circulant	O
Hadamard	O
matrix	B-Architecture
,	O
</s>
<s>
a	O
single	O
point	O
-	O
one	O
of	O
the	O
Butson-type	B-Algorithm
Hadamard	I-Algorithm
matrices	I-Algorithm
,	O
.	O
</s>
<s>
It	O
is	O
not	O
known	O
,	O
however	O
,	O
if	O
this	O
list	O
is	O
complete	O
,	O
but	O
it	O
is	O
conjectured	O
that	O
is	O
an	O
exhaustive	O
(	O
but	O
not	O
necessarily	O
irredundant	O
)	O
list	O
of	O
all	O
complex	B-Algorithm
Hadamard	I-Algorithm
matrices	I-Algorithm
of	O
order	O
6	O
.	O
</s>
