So I'm writing an assignment in which I have to define, what is understood by a hermitian operator.

My teacher has told me to definere it as:

<ϕm|A|ϕn> = <ϕn|A|ϕm>* , where lϕn> and lϕm> is the n'th and m'th basis vector.

And using this i then have to proof the more general definition:

<a|A|b> = <b|A|a>*, la> and lb> being arbitrary vectors.

I've tried to do so but I have yet not succeeded:

What I've done is to say: Take a hermitian operator. Since it's hermitian it must satisfy:

A = ∑_(m,n)|ϕm>amn <ϕn| = ∑(m,n)|ϕm> anm*<ϕm|

Which when dotted with 2 arbitrary vectors|ψ> and <φ|equals to:

<φ|A|ψ> = ∑(m,n) <φ|ϕm>amn<ϕn|ψ> = ∑(m,n) <ϕm|φ>*amn <ψ|ϕn>* =

∑(m,n) <ψ|ϕn>* amn <ϕm|φ>*

Since A is hermitian this equals to:

∑(m,n) <ψ|ϕn>* amn <ϕm|φ>* = ∑(m,n) <ψ|ϕn>*anm*<ϕm|φ>* = [∑(m,n) <ψ|ϕn>anm<ϕm|φ>]*

Now the proof would work if |ϕn>anm<ϕm| = |ϕm>amn<ϕn|, but that's not right is it?

Can anyone help me how to proove this? I know it's simple, but I'm still finding it a little hard.