Vectors and Matrices | Part IA, 2008

Let UU be an n×nn \times n unitary matrix (UU=UU=I)\left(U^{\dagger} U=U U^{\dagger}=I\right). Suppose that AA and BB are n×nn \times n Hermitian matrices such that U=A+iBU=A+i B.

Show that

(i) AA and BB commute,

(ii) A2+B2=IA^{2}+B^{2}=I.

Find AA and BB in terms of UU and UU^{\dagger}, and hence show that AA and BB are uniquely determined for a given UU.

Typos? Please submit corrections to this page on GitHub.