A special unitary matrix is an 
square matrix 
with complex
entries satisfying
 |
(1)
|
where 
is a positive integer, 
is the identity matrix,
is the conjugate transpose of 
, and the determinant is
 |
(2)
|
The first condition means that 
is a unitary matrix, and
the second condition provides a restriction beyond a general unitary
matrix, which may have determinant 
for 
any real number. For example,
![1/(sqrt(2))[i i; i -i]](/images/equations/SpecialUnitaryMatrix/NumberedEquation3.svg) |
(3)
|
is a special unitary matrix.
Its matrix inverse is 
. Its column vectors and
row vectors form orthonormal
bases of the complex vector space 
for the standard Hermitian
inner product. A real matrix is special unitary
iff it is a special
orthogonal matrix.
A 
matrix is special unitary iff it
has the form
![U=[a b; -b^_ a^_],](/images/equations/SpecialUnitaryMatrix/NumberedEquation4.svg) |
(4)
|
where 
,
,
and the bars denote complex conjugates (Ng et
al. 2021, pp. 141-143).
The 
special unitary matrices are closed under matrix
multiplication and the inverse operation, and
therefore form a matrix group called the special
unitary group 
.
