Number of eigenvalues of this Hermitian

In summary, the student is trying to solve a problem from a book on quantum mechanics. He is having difficulty understanding why there are three directions in which angular velocity and angular momentum are parallel. He has found an equation that states that l and ω will be parallel if ω is an eigenvector of l. However, he still does not understand how to prove that this is true. He is looking for help with this problem.
  • #1
kuchenvater
9
0
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?
 
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  • #2
That would be correct if you prove that your matrix M has three distinct eigenvalues. That is not always the case. Consider a set of 8 masses placed on the corners of a cube, with origin at the center of the cube. The moment of inertia tensor M for this is a diagonal matrix with all diagonal elements equal to one another. There's only one eigenvalue.

What you need to show is that one can always at least three directions along which angular velocity and angular momentum are parallel, regardless of the number of unique eigenvalues.

Aside: A eigenvalue of zero is a bit problematic. This means zero angular momentum for rotation along that eigenvector. You'd have to interpret the zero vector as being parallel to all vectors.
 
  • #3
D H said:
What you need to show is that one can always at least three directions along which angular velocity and angular momentum are parallel, regardless of the number of unique eigenvalues.
Any hints on how to do that?
 
  • #4
The matrix is Hermitian. What does the spectral theorem say about the eigenvectors?
 
  • #5


Yes, your reasoning is correct. Since M is a Hermitian matrix, it will have real eigenvalues. Additionally, since it describes a rotating system of mass, it must be defined in the real plane and have 3 axes (x, y, and z). This means that the matrix will have 3 distinct eigenvalues, corresponding to each of the axes. Therefore, there are 3 eigenvalues of M, and as you mentioned, they can be found by considering the diagonal elements of the diagonalized form of M.
 

1. What is a Hermitian matrix?

A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. This means that if A is a Hermitian matrix, then A* = A, where A* is the conjugate transpose of A. In simpler terms, a Hermitian matrix is a square matrix with complex entries that is symmetric about its main diagonal.

2. What is the significance of a Hermitian matrix?

Hermitian matrices have many important properties and applications in mathematics and science. One of the main reasons they are studied is because they have real eigenvalues and orthogonal eigenvectors, making them useful for diagonalization and solving systems of linear equations. They also have applications in quantum mechanics, signal processing, and data analysis.

3. How many eigenvalues does a Hermitian matrix have?

A Hermitian matrix always has the same number of eigenvalues as its size. For example, a 3x3 Hermitian matrix will have 3 eigenvalues, and a 5x5 Hermitian matrix will have 5 eigenvalues. This is because the eigenvalues correspond to the solutions of the characteristic equation, which has the same degree as the matrix.

4. Can a Hermitian matrix have complex eigenvalues?

No, a Hermitian matrix can only have real eigenvalues. This is because the eigenvalues of a Hermitian matrix are the same as the eigenvalues of its conjugate transpose, and the eigenvalues of a conjugate transpose are the complex conjugates of the eigenvalues of the original matrix. Since a Hermitian matrix is equal to its own conjugate transpose, its eigenvalues must be equal to their complex conjugates, making them real.

5. How can I determine the number of eigenvalues of a Hermitian matrix?

To determine the number of eigenvalues of a Hermitian matrix, you can simply count the number of entries in the matrix. Alternatively, you can use the characteristic equation to find the eigenvalues, as the degree of the equation will give you the number of eigenvalues. Additionally, you can use properties of Hermitian matrices, such as the fact that they have real eigenvalues, to determine the number of eigenvalues without explicitly finding them.

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