How Many Real Variables Are Needed to Construct a 2x2 Unitary Matrix?

[gc2004]

Homework Statement

how many real variables would be required to construct a most general 2by2 unitary matrix?

Homework Equations

a unitary matrix U is one for which the U(U hermitian) = identity matrix or (U hermitian)U = identity matrix

The Attempt at a Solution

first i wrote the unitary matrix as {{a1+ib1,a2+ib2},{a3+ib3,a4+ib4}}, where 'i' is square root of -1. using the condition U(U hermitian) = identity matrix, i get four independent equations. thus, i should expect the number of real variables required as 8 - 4 (number of constraints) = 4. but if i consider the definition (U hermitian)U = identity matrix, i get another set of four equations. Does that mean that the number of real variables required is 8-8 =0?

there is another way to attack the problem. a hermitian matrix must have real diagonal elements. the (1,2) element must be the complex conjugate of the (2,1) element and hence, i need 4 real variables to construct the most general 2by2 hermitian matrix. since a unitary matrix can be written as exp(iH) where H is a hermitian matrix, does this also indicate that i would need 4 variables for a unitary matrix as well?

 

[Dick]

You are counting correctly, except (U*)U=I and U(U*)=I are not independent sets of equations. If one holds the other automatically holds. There are 4 real parameters describing a 2x2 unitary matrix.

 

[gc2004]

thank you for your assistance... i later on figured out that it must be true since when i impose the condition that the determinant of the matrix is 1, then i end up getting SU2 group, to specify which i need three parameters... so, without the constraint of the determinant being 1, i must need 4 parameters