<s>
Transpositions	B-Algorithm
matrix	I-Algorithm
(	O
Tr	O
matrix	B-Architecture
)	O
is	O
square	O
matrix	B-Architecture
,	O
,	O
,	O
which	O
elements	O
are	O
obtained	O
from	O
the	O
elements	O
of	O
given	O
n-dimensional	O
vector	O
as	O
follows	O
:	O
,	O
where	O
denotes	O
operation	O
"	O
bitwise	O
Exclusive	O
or	O
"	O
(	O
XOR	O
)	O
.	O
</s>
<s>
matrix	B-Architecture
is	O
symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
matrix	B-Architecture
is	O
persymmetric	B-Algorithm
matrix	I-Algorithm
,	O
i.e.	O
</s>
<s>
it	O
is	O
symmetric	B-Algorithm
with	O
respect	O
to	O
the	O
northeast-to-southwest	O
diagonal	O
too	O
.	O
</s>
<s>
Every	O
one	O
row	O
and	O
column	O
of	O
matrix	B-Architecture
consists	O
all	O
n	O
elements	O
of	O
given	O
vector	O
without	O
repetition	O
.	O
</s>
<s>
Every	O
two	O
rows	O
matrix	B-Architecture
consists	O
fours	O
of	O
elements	O
with	O
the	O
same	O
values	O
of	O
the	O
diagonal	O
elements	O
.	O
</s>
<s>
In	O
example	O
if	O
and	O
are	O
two	O
arbitrary	O
selected	O
elements	O
from	O
the	O
same	O
column	O
q	O
of	O
matrix	B-Architecture
,	O
then	O
,	O
matrix	B-Architecture
consists	O
one	O
fours	O
of	O
elements	O
,	O
for	O
which	O
are	O
satisfied	O
the	O
equations	O
and	O
.	O
</s>
<s>
The	O
figure	O
on	O
the	O
right	O
shows	O
some	O
fours	O
of	O
elements	O
in	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
property	O
of	O
fours	O
of	O
matrices	O
gives	O
the	O
possibility	O
to	O
create	O
matrix	B-Architecture
with	O
mutually	O
orthogonal	O
rows	O
and	O
columns	O
(	O
matrix	B-Architecture
)	O
by	O
changing	O
the	O
sign	O
to	O
an	O
odd	O
number	O
of	O
elements	O
in	O
every	O
one	O
of	O
fours	O
,	O
.	O
</s>
<s>
In	O
 [ 5 ] 	O
is	O
offered	O
algorithm	O
for	O
creating	O
matrix	B-Architecture
using	O
Hadamard	O
product	O
,	O
(	O
denoted	O
by	O
)	O
of	O
Tr	O
matrix	B-Architecture
and	O
n-dimensional	O
Hadamard	O
matrix	B-Architecture
whose	O
rows	O
(	O
except	O
the	O
first	O
one	O
)	O
are	O
rearranged	O
relative	O
to	O
the	O
rows	O
of	O
Sylvester-Hadamard	O
matrix	B-Architecture
in	O
order	O
,	O
for	O
which	O
the	O
rows	O
of	O
the	O
resulting	O
Trs	O
matrix	B-Architecture
are	O
mutually	O
orthogonal	O
.	O
</s>
<s>
is	O
n-dimensional	O
Identity	O
matrix	B-Architecture
.	O
</s>
<s>
is	O
n-dimensional	O
Hadamard	O
matrix	B-Architecture
,	O
which	O
rows	O
are	O
interchanged	O
against	O
the	O
Sylvester-Hadamard[4]	O
matrix	B-Architecture
in	O
given	O
order	O
for	O
which	O
the	O
rows	O
of	O
the	O
resulting	O
matrix	B-Architecture
are	O
mutually	O
orthogonal	O
.	O
</s>
<s>
is	O
the	O
vector	O
from	O
which	O
the	O
elements	O
of	O
matrix	B-Architecture
are	O
derived	O
.	O
</s>
<s>
Orderings	O
R	O
of	O
Hadamard	O
matrix	B-Architecture
’s	O
rows	O
were	O
obtained	O
experimentally	O
for	O
matrices	O
of	O
sizes	O
2	O
,	O
4	O
and	O
8	O
.	O
</s>
<s>
It	O
is	O
important	O
to	O
note	O
,	O
that	O
the	O
ordering	O
R	O
of	O
Hadamard	O
matrix	B-Architecture
’s	O
rows	O
(	O
against	O
the	O
Sylvester-Hadamard	O
matrix	B-Architecture
)	O
does	O
not	O
depend	O
on	O
the	O
vector	O
.	O
</s>
<s>
)	O
,	O
then	O
matrix	B-Architecture
(	O
obtained	O
as	O
it	O
was	O
described	O
above	O
)	O
is	O
matrix	B-Architecture
of	O
reflection	B-Algorithm
.	O
</s>
<s>
Transpositions	B-Algorithm
matrix	I-Algorithm
with	O
mutually	O
orthogonal	O
rows	O
(	O
matrix	B-Architecture
)	O
of	O
order	O
4	O
for	O
vector	O
is	O
obtained	O
as	O
:	O
</s>
<s>
where	O
is	O
matrix	B-Architecture
,	O
obtained	O
from	O
vector	O
,	O
and	O
""	O
denotes	O
operation	O
Hadamard	O
product	O
and	O
is	O
Hadamard	O
matrix	B-Architecture
,	O
which	O
rows	O
are	O
interchanged	O
in	O
given	O
order	O
for	O
which	O
the	O
rows	O
of	O
the	O
resulting	O
matrix	B-Architecture
are	O
mutually	O
orthogonal	O
.	O
</s>
<s>
As	O
can	O
be	O
seen	O
from	O
the	O
figure	O
above	O
,	O
the	O
first	O
row	O
of	O
the	O
resulting	O
matrix	B-Architecture
contains	O
the	O
elements	O
of	O
the	O
vector	O
without	O
transpositions	O
and	O
sign	O
change	O
.	O
</s>
