[AntonioLao]

An operator or matrix U is unitary if UU+=U+U=I where U+ is the Hermitian conjugate or conjugate transpose of U and I is the identity matrix with trace equals to the order of the matrix. Furthermore, the rows or columns of the matrix form an orthonormal set. Orthonormality, in this case, means that the scalar dot product of one row with itself is unity while the scalar dot product of one row with any other row is identically zero. The eigenvalues of a unitary matrix also have unit magnitude: |l|²=1 where l is an eigenvalue. Its determinant is also unity.

Spin½ particle wavefunctions have two components: y¹ and y² forming a spinor. These can undergo linear rotation given by: y¹’=ay¹+by² and y²’=cy¹+dy². The factors: a, b, c, d are in general complex functions of the angles of rotation of the coordinate axes. Given 2 spinors y and f their bilinear expression undergoes rotation into itself by y¹’f²’-y²’f¹’=(ad-bc)(y¹f²-y²f¹) where a, b, c, d now become the elements of a transformation matrix U. If only one function is transformed into itself then it corresponds to zero spin such that ad-bc=1 with transformation determinant of unity. Therefore, U is a unitary matrix and its inverse exists which is equal to its Hermitian conjugate matrix. The four complex functions a, b, c, d at the most contain only 3 independent parameters. These correspond to the 3 directional cosines defining a rotation of a 3-dimension coordinate system.