## Properties of the Eigenvalues and Eigenvectors of a Hermitian Operator

# Properties of the Eigenvalues and Eigenvectors of a Hermitian Operator

We can say that an operator *A** *is a Hermitian Operator if it satisfies that: .

Given the eigenvalues and eigenvectors of the Hermitian Operators, the following properties apply:

*The eigenvalues of a Hermitian operator are real.**Two eigenvectors of a Hermitian operator corresponding to two different eigenvalues are orthogonal.*

In this section,we are to prove these statements.

**Proof:**

**Proof:**

*(1.)* *The eigenvalues of a Hermitian operator are real.*

Suppose that we have a Hermitian operator , then we have

[1]

where is the eigen vector and is the eigenvalue of the operator .

Taking the scalar product of [1] with yields,

[2]

Here, we have to be **real*** *because we assumed our operator to be a Hermitian operator already. And hence this implies that is also **real***.*

Now, we note that is a Hermitian operator and by definition, .

Then for the conjugate we can have,

[3]

Since the definition establishes that , then we equate the scalar products of these 2 operators.

[4]

Using equations [2] and [3], we have

[5]

[6]

We know already that . So we are left with,

Since we have seen that the eigenvalues of the conjugate of the Hermitian operator to be * real*.

Therefore we can say that the *eigenvalues of a Hermitian operator are real. *

*(2.) **Two eigenvectors of a Hermitian operator corresponding to two different eigenvalues are orthogonal. *

Let and be eigenvectors of a Hermitian operator , corresponding to the eigenvalues and respectively.

For both eigenvectors to be orthogonal, it must satisfy that: .

Now, we have

[7]

[8]

Since is Hermitian, we can write [8] in the form

[9]

If we take the scalar product of [7] with and [9] with , then

[10]

[11]

Equating equations [10] and [11] yields.

[12]

[13]

[14]

Since , then the only way for equation [14] to be true is to set .

Thus, *two eigenvectors of a Hermitian operator corresponding to two different eigenvalues are orthogonal. *

**By: Joshua J. Ordeniza, MS Physics Student, MSU-IIT**