Degenerate Eigenvalues and Eigenvectors: Understanding Differences in Solutions

In summary, the conversation is discussing the solution to a problem involving finding the eigenvalues and eigenvectors of a 4x4 matrix. The first line finds the eigenvalues of the matrix, while the second line finds the eigenvectors. The third and fourth lines take different combinations of rows from the matrix to find the determinant and the conversation discusses whether this is a valid technique. The conclusion is that while the eigenvalues in the first and last cases are the same, the eigenvectors are different due to the degeneracy of the eigenvalues.
  • #1
Nusc
760
2

Homework Statement


Please see the attached image.

The first line just finds the eigenvalues of that matrix.

The second line finds the eigenvectors.

The third line just takes row 1 and row 3 of that matrix and find the determinant.
The fourth line just takes row 2 and row 4 of that matrix and find the determinant.


Because the two sets of equations are identitical, the eigenval
ues are double degenerate in the later case. Thus the evectors are not fixed.

But in the former case, the eigenvalues/eigenvecotrs are different.


THe solution is the later but I don't understand why the former part gives different answers.

What's wrong?

Homework Equations





The Attempt at a Solution

 

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  • #2
I'm not sure what you're getting at. You're getting the same set of eigenvalues whether you use the 4x4 matrix or the two 2x2 matrices. Are you asking why you have more freedom to choose the eigenvectors in the 4x4 case?
 
  • #3
No that's the thing. The eigenvalues are not the same.

Is the technique on line 3 and 4 valid? the equations are independent of each other.
 
  • #4
You have the same set of four eigenvalues. How are the two sets different?
 
  • #5
In the first line, there were 4 distinct eigenvalues.In the third & fourth line there are 2 degenerate eigenvalues.

Do you know what I mean?

Line 3 examines row 1 and row 3 in the matrix and takes the determinant of that seperately from

Line 4 which examines row 2 and row 4 in the matrix and takes the determinant of that.

Can you do this? Clearly, they're not equivalent.
 
  • #6
Nusc said:
In the first line, there were 4 distinct eigenvalues.

How exactly do you consider [tex]-\sqrt{3}\sqrt{3A^2+4B^2}[/itex] to be distinct from [tex]-\sqrt{3}\sqrt{3A^2+4B^2}[/itex]? :confused:

In your first line, you do not have 4 distinct eigenvalues; you have two doubly degenerate eigenvalues.
 
  • #7
Those values each appear twice among the four eigenvalues in both cases. How are they different?
 
  • #8
Haha you're right. I took the determinant by hand and solved it and didn't see what was written in mathematica...
 

1. What are degenerate eigenvalues?

Degenerate eigenvalues are eigenvalues that have more than one corresponding eigenvector. This means that there are multiple linearly independent solutions to the eigenvalue equation for the same eigenvalue.

2. How do degenerate eigenvalues affect matrix diagonalization?

Degenerate eigenvalues can complicate matrix diagonalization because they require finding a complete set of linearly independent eigenvectors for each degenerate eigenvalue. This can lead to a larger and more complex diagonalization process.

3. Can degenerate eigenvalues be negative?

Yes, degenerate eigenvalues can be negative. The degeneracy of an eigenvalue is not related to its sign, but rather to the number of linearly independent eigenvectors associated with it.

4. What are the physical implications of degenerate eigenvalues?

Degenerate eigenvalues have important physical implications in quantum mechanics, where they often represent the energy levels of a system. In systems with degenerate eigenvalues, there can be multiple possible states with the same energy, leading to complex behaviors and phenomena.

5. How are degenerate eigenvalues handled in practical applications?

In practical applications, degenerate eigenvalues are handled by finding a complete set of linearly independent eigenvectors for each degenerate eigenvalue. This allows for a complete description of the system and its behavior, and is crucial in fields such as quantum mechanics and signal processing.

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