Hermitian operator proof

[Lchan1]

Homework Statement

A Hermitian operator is such that, for arbitrary vectors ai and aj
in a vector space,we have (ai,Oaj) = (Oai, aj).
Prove that if for an arbitrary vector a in the vector space, the operator O satisfies
(a,Oa) = (Oa, a),
then O is Hermitian

Homework Equations

The Attempt at a Solution

Our prof gave us a hint:
let a=a1+z1a2 and a=a1+z2a2. and find the appropriate zs to satisfy the proof. I tried doing the inner product using the different a. and got myself no where.

 

[mighty2000]

Dear student,

Thank you for your post. It is great to see that you are seeking help with your work. I am a scientist and I would be happy to assist you in solving this problem. Let's start by defining a Hermitian operator. A Hermitian operator is a linear operator that satisfies the condition (ai,Oaj) = (Oai, aj) for all arbitrary vectors ai and aj in a vector space. This means that the operator O is symmetric with respect to the inner product of the vectors ai and aj. In other words, the order of the vectors in the inner product does not matter when the operator O is applied.

Now, let us consider the given condition (a,Oa) = (Oa, a) for an arbitrary vector a in the vector space. We want to prove that this condition implies that the operator O is Hermitian. To do this, we can use the hint given by your professor. Let us define two vectors a1 and a2 such that a = a1 + za2. Substituting this into the given condition, we get:

(a,Oa) = (Oa, a)
(a,a1 + za2) = (Oa, a1 + za2)
(a,a1) + z(a,a2) = (Oa, a1) + z(Oa, a2)

Now, let us define another vector a' = a1 + z'a2, where z' is a different value of z. Substituting this into the condition, we get:

(a',Oa') = (Oa', a')
(a',a1 + z'a2) = (Oa', a1 + z'a2)
(a',a1) + z'(a',a2) = (Oa', a1) + z'(Oa', a2)

Since a' and a are arbitrary vectors, we can equate the terms in the above two equations to get:

(a,a1) + z(a,a2) = (a',a1) + z'(a',a2)
(a,a1) = (a',a1)
z(a,a2) = z'(a',a2)

Since z and z' are different values, this can only be true if (a,a2) = (a',a2). This implies that the operator O is symmetric with respect to the inner product of the vectors a and