<s>
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
real	O
analogue	O
of	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
is	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Unitary	B-Algorithm
matrices	I-Algorithm
have	O
significant	O
importance	O
in	O
quantum	O
mechanics	O
because	O
they	O
preserve	O
norms	O
,	O
and	O
thus	O
,	O
probability	O
amplitudes	O
.	O
</s>
<s>
For	O
any	O
unitary	B-Algorithm
matrix	I-Algorithm
of	O
finite	O
size	O
,	O
the	O
following	O
hold	O
:	O
</s>
<s>
is	O
normal	B-Algorithm
(	O
)	O
.	O
</s>
<s>
is	O
diagonalizable	B-Algorithm
;	O
that	O
is	O
,	O
is	O
unitarily	B-Algorithm
similar	I-Algorithm
to	O
a	O
diagonal	O
matrix	O
,	O
as	O
a	O
consequence	O
of	O
the	O
spectral	O
theorem	O
.	O
</s>
<s>
can	O
be	O
written	O
as	O
,	O
where	O
indicates	O
the	O
matrix	O
exponential	O
,	O
is	O
the	O
imaginary	O
unit	O
,	O
and	O
is	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
For	O
any	O
nonnegative	O
integer	O
,	O
the	O
set	O
of	O
all	O
unitary	B-Algorithm
matrices	I-Algorithm
with	O
matrix	O
multiplication	O
forms	O
a	O
group	O
,	O
called	O
the	O
unitary	O
group	O
.	O
</s>
<s>
Any	O
square	B-Architecture
matrix	I-Architecture
with	O
unit	O
Euclidean	O
norm	O
is	O
the	O
average	O
of	O
two	O
unitary	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
is	O
a	O
normal	B-Algorithm
matrix	I-Algorithm
(	O
equivalently	O
,	O
there	O
is	O
an	O
orthonormal	O
basis	O
formed	O
by	O
eigenvectors	O
of	O
)	O
with	O
eigenvalues	O
lying	O
on	O
the	O
unit	O
circle	O
.	O
</s>
<s>
This	O
expression	O
highlights	O
the	O
relation	O
between	O
unitary	B-Algorithm
matrices	I-Algorithm
and	O
orthogonal	B-Algorithm
matrices	I-Algorithm
of	O
angle	O
.	O
</s>
<s>
Many	O
other	O
factorizations	O
of	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
in	O
basic	O
matrices	O
are	O
possible	O
.	O
</s>
