Special Unitary Matrix

A square matrix is a special unitary matrix if

(1)

where is the identity matrix and is the conjugate transpose matrix, 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,

(3)

is a special unitary matrix. A matrix can be tested to see if it is a special unitary matrix using the Wolfram Language function

  SpecialUnitaryQ[m_List?MatrixQ] :=
    (Conjugate @ Transpose @ m . m ==
      IdentityMatrix @ Length @ m&& Det[m] == 1)

The special unitary matrices are closed under multiplication and the inverse operation, and therefore form a matrix group called the special unitary group .