# Unitary matrix problem

## The Attempt at a Solution

I satrted this as U=adjoint of AB
u_ik=sum(j)[(a_ij)*(b_jk)]
I know then,I may take tarnspose of both sides so that we have:[(u_ki)~]=sum(j){[(b_kj)~][(a_ji)~]}
then [U~]=[B~][A~]
Then,we are done.But,this proves for real elements.I am not sure that this proves also for imaginary elements...

nrqed
Homework Helper
Gold Member

## The Attempt at a Solution

I satrted this as U=adjoint of AB
u_ik=sum(j)[(a_ij)*(b_jk)]
I know then,I may take tarnspose of both sides so that we have:[(u_ki)~]=sum(j){[(b_kj)~][(a_ji)~]}
then [U~]=[B~][A~]
Then,we are done.But,this proves for real elements.I am not sure that this proves also for imaginary elements...

The proof works exactlythe same way, you just have to include a complex conjugate of the elements when you take the adjoint. Taking the complex conjugate does not change anything to the indices, so the proof still works...you just have complex conjugates everywhere.

OK,I thought of this possibility as I am not using any extra property of complex matrices.
Do I need to write (a_ij)* in those cases?

Dick
Homework Helper
I think you guys are making this too complicated. By definition: