# Unitary matrix problem

• Kolahal Bhattacharya
In summary, the conversation discusses the proof that the adjoint of the product of two matrices is equal to the product of their adjoints. The proof works for both real and imaginary elements, as taking the complex conjugate does not change the indices. The definition of the adjoint is also mentioned as a simpler way to understand the proof.

## 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...

Kolahal Bhattacharya said:

## 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?

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