Number of eigenvalues of this Hermitian

Hi. I'm trying to study QM from Shankar on my own. Asking this here because I don't really have a teacher to help me with this:

Homework Statement

I'm trying to solve problem 1.8.9 -part 3 of "The Principles of Quantum Mechanics" by R Shankar. Here's the problem:
Given the values of Mij (see next section for equations) which is Hermitian, argue that there exist three directions such that the angular velocity and angular momentum will be parallel.

Homework Equations

Given
Mij=∑a=ma[ r2a δij - (ra)i(ra)j ]
here i denotes the ith Cartesian component.

|l>=M|ω>

The Attempt at a Solution

It's known that M is Hermitian. l and ω will be parallel if ω is an eigenvector of l. So I have to prove that the number of eigenvalues of M is 3. Somehow I can't figure out a way to do that. It's obvious that all eigenvalues will be real and the number of eigenvalues is equal to or less than the rank of a matrix. I was thinking I could consider only the diagonal elements of a diagonalized version of M or just the trace of M(which yields the sum of all eigenvalues) by setting i=j and then I'd have i as the index of sigma. If I can count the number of values of i, I'd have the number of eigenvalues. And since M is Hermitian and has real eigenvalues and this describes a rotating system of mass(i.e. it's moment of inertia), it must be defined in the real plane and that has 3 axes-x,y and z, one for each value of i. So i varies across 3 values, hence the diagonalized matrix has 3 diagonal elements and hence 3 eigenvalues. The question also mentions the mass's cartesian component, so that automatically means x, y, z.

Is that correct reasoning?

D H
Staff Emeritus