AIMS Lie Groups: Fall-2022
Practice Exercises 3 : Due 03 Wed 7 Dec

1. Unitary matrices S0 4483S Unitary matrices are complex matrices that satisfy the condition \(M^\dagger M=I\), where \(I\) denotes the identity matrix. Special unitary matrices satisfy the additional condition that \(\det M=1\). The \(n\times n\) special unitary matrices are denoted by \(SU(n)\), which turns out to be a Lie group.
   1. Find any one element of \(SU(2)\).
   2. Find at least one 1-parameter family of elements of \(SU(2)\), that is, a family of matrices \(M(\alpha)\in SU(2)\) satisfying:
      • \(M(0)=I\)
      • \(M(\alpha+\beta)=M(\alpha)M(\beta)\)
   3. Find the most general element of \(SU(2)\).