<s>
In	O
mathematics	O
,	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
,	O
also	O
known	O
as	O
the	O
Hermitian	B-Algorithm
transpose	I-Algorithm
,	O
of	O
an	O
complex	O
matrix	B-Architecture
is	O
an	O
matrix	B-Architecture
obtained	O
by	O
transposing	O
and	O
applying	O
complex	O
conjugate	O
on	O
each	O
entry	O
(	O
the	O
complex	O
conjugate	O
of	O
being	O
,	O
for	O
real	O
numbers	O
and	O
)	O
.	O
</s>
<s>
For	O
real	B-Architecture
matrices	I-Architecture
,	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
is	O
just	O
the	O
transpose	O
,	O
.	O
</s>
<s>
where	O
denotes	O
the	O
transpose	O
and	O
denotes	O
the	O
matrix	B-Architecture
with	O
complex	O
conjugated	O
entries	O
.	O
</s>
<s>
Other	O
names	O
for	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
a	O
matrix	B-Architecture
are	O
Hermitian	B-Algorithm
conjugate	O
,	O
adjoint	B-Algorithm
matrix	I-Algorithm
or	O
transjugate	O
.	O
</s>
<s>
The	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
a	O
matrix	B-Architecture
can	O
be	O
denoted	O
by	O
any	O
of	O
these	O
symbols	O
:	O
</s>
<s>
In	O
some	O
contexts	O
,	O
denotes	O
the	O
matrix	B-Architecture
with	O
only	O
complex	O
conjugated	O
entries	O
and	O
no	O
transposition	O
.	O
</s>
<s>
Suppose	O
we	O
want	O
to	O
calculate	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
the	O
following	O
matrix	B-Architecture
.	O
</s>
<s>
We	O
first	O
transpose	O
the	O
matrix	B-Architecture
:	O
</s>
<s>
Then	O
we	O
conjugate	O
every	O
entry	O
of	O
the	O
matrix	B-Architecture
:	O
</s>
<s>
Hermitian	B-Algorithm
or	O
self-adjoint	O
if	O
;	O
i.e.	O
,	O
.	O
</s>
<s>
Skew	O
Hermitian	B-Algorithm
or	O
antihermitian	O
if	O
;	O
i.e.	O
,	O
.	O
</s>
<s>
Normal	B-Algorithm
if	O
.	O
</s>
<s>
Unitary	B-Algorithm
if	O
,	O
equivalently	O
,	O
equivalently	O
.	O
</s>
<s>
Even	O
if	O
is	O
not	O
square	O
,	O
the	O
two	O
matrices	O
and	O
are	O
both	O
Hermitian	B-Algorithm
and	O
in	O
fact	O
positive	B-Algorithm
semi-definite	I-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
The	O
conjugate	B-Algorithm
transpose	I-Algorithm
"	O
adjoint	O
"	O
matrix	B-Architecture
should	O
not	O
be	O
confused	O
with	O
the	O
adjugate	O
,	O
,	O
which	O
is	O
also	O
sometimes	O
called	O
adjoint	O
.	O
</s>
<s>
The	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
a	O
matrix	B-Architecture
with	O
real	O
entries	O
reduces	O
to	O
the	O
transpose	O
of	O
,	O
as	O
the	O
conjugate	O
of	O
a	O
real	O
number	O
is	O
the	O
number	O
itself	O
.	O
</s>
<s>
The	O
conjugate	B-Algorithm
transpose	I-Algorithm
can	O
be	O
motivated	O
by	O
noting	O
that	O
complex	O
numbers	O
can	O
be	O
usefully	O
represented	O
by	O
real	B-Architecture
matrices	I-Architecture
,	O
obeying	O
matrix	B-Architecture
addition	O
and	O
multiplication	O
:	O
</s>
<s>
That	O
is	O
,	O
denoting	O
each	O
complex	O
number	O
by	O
the	O
real	O
matrix	B-Architecture
of	O
the	O
linear	B-Architecture
transformation	I-Architecture
on	O
the	O
Argand	O
diagram	O
(	O
viewed	O
as	O
the	O
real	O
vector	O
space	O
)	O
,	O
affected	O
by	O
complex	O
-multiplication	O
on	O
.	O
</s>
<s>
Thus	O
,	O
an	O
matrix	B-Architecture
of	O
complex	O
numbers	O
could	O
be	O
well	O
represented	O
by	O
a	O
matrix	B-Architecture
of	O
real	O
numbers	O
.	O
</s>
<s>
The	O
conjugate	B-Algorithm
transpose	I-Algorithm
,	O
therefore	O
,	O
arises	O
very	O
naturally	O
as	O
the	O
result	O
of	O
simply	O
transposing	O
such	O
a	O
matrix	B-Architecture
—	O
when	O
viewed	O
back	O
again	O
as	O
an	O
matrix	B-Architecture
made	O
up	O
of	O
complex	O
numbers	O
.	O
</s>
<s>
for	O
any	O
complex	O
number	O
and	O
any	O
matrix	B-Architecture
.	O
</s>
<s>
for	O
any	O
matrix	B-Architecture
and	O
any	O
matrix	B-Architecture
.	O
</s>
<s>
for	O
any	O
matrix	B-Architecture
,	O
i.e.	O
</s>
<s>
Hermitian	B-Algorithm
transposition	O
is	O
an	O
involution	B-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
square	O
matrix	B-Architecture
,	O
then	O
where	O
denotes	O
the	O
determinant	O
of	O
.	O
</s>
<s>
If	O
is	O
a	O
square	O
matrix	B-Architecture
,	O
then	O
where	O
denotes	O
the	O
trace	O
of	O
.	O
</s>
<s>
for	O
any	O
matrix	B-Architecture
,	O
any	O
vector	O
in	O
and	O
any	O
vector	O
.	O
</s>
<s>
The	O
last	O
property	O
given	O
above	O
shows	O
that	O
if	O
one	O
views	O
as	O
a	O
linear	B-Architecture
transformation	I-Architecture
from	O
Hilbert	O
space	O
to	O
then	O
the	O
matrix	B-Architecture
corresponds	O
to	O
the	O
adjoint	O
operator	O
of	O
.	O
</s>
<s>
The	O
concept	O
of	O
adjoint	O
operators	O
between	O
Hilbert	O
spaces	O
can	O
thus	O
be	O
seen	O
as	O
a	O
generalization	O
of	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
matrices	O
with	O
respect	O
to	O
an	O
orthonormal	O
basis	O
.	O
</s>
<s>
Another	O
generalization	O
is	O
available	O
:	O
suppose	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
a	O
complex	O
vector	O
space	O
to	O
another	O
,	O
,	O
then	O
the	O
complex	O
conjugate	O
linear	B-Architecture
map	I-Architecture
as	O
well	O
as	O
the	O
transposed	B-Algorithm
linear	I-Algorithm
map	I-Algorithm
are	O
defined	O
,	O
and	O
we	O
may	O
thus	O
take	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
to	O
be	O
the	O
complex	O
conjugate	O
of	O
the	O
transpose	O
of	O
.	O
</s>
