Thread: Unitary matrices- General form

Does anybody know the general form of a 2x2 unitary matrix such that GG*=I. Where I is identity matrix and G* is conjugate traspose of G. I have been told that it is:

a -b*
b a*

Is this correct???

Originally Posted by johnbarkwith

Does anybody know the general form of a 2x2 unitary matrix such that GG*=I. Where I is identity matrix and G* is conjugate traspose of G. I have been told that it is:

a -b*
b a*

Is this correct???

That is partially correct. To have that form, the matrix must be in the special unitary group SU(2). This means that in addition to being unitary, the matrix has determinant 1.

If the matrix is unitary then . Compare matrix elements on both sides of that equation, together with the equation ad-bc=1, and you will find that and . (I'm using bars rather than stars to denote complex conjugates.)

The general unitary matrix could have determinant –1, in which case it has the form .

Originally Posted by Opalg

The general unitary matrix could have determinant –1, in which case it has the form .

Actually.... a general unitary matrix can have determinant any complex number of absolute value 1. So the matrix
for is also unitary, with determinant . (It is also necessary that .)