Multiplicity of Eigenvalues in a 3x3 Matrix

In summary, the matrix A has two real eigenvalues, -1 and 0, and a basis of each eigenspace. The equation -\lambda^3 - \lambda^2 = 0 has one real eigenvalue of multiplicity 2 and one of multiplicity 1.
  • #1
icefall5
19
0

Homework Statement


The matrix [itex]A = \begin {bmatrix} 0 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & 1 & 0 \end{bmatrix}[/itex] has two real eigenvalues, one of multiplicity 2 and one of multiplicity 1. Find the eigenvalues and a basis of each eigenspace.

Homework Equations


N/A

The Attempt at a Solution


I've done cofactor expansion to come up with the equation [itex] - \lambda^3 - \lambda^2[/itex], and that the eigenvalues are therefore -1 and 0, but I don't know how to determine the multiplicity of each. I've looked it up and have gotten nowhere. I can do the rest of the problem, I just don't know how to get these multiplicities.

EDIT: Working on the problem further, and I got the basis for -1. There are supposed to be two vectors for 0, however, but the RREF of the matrix is [itex]\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}[/itex]. How does that translate to two eigenvectors?

Thanks in advance for any help!
 
Last edited:
Physics news on Phys.org
  • #2
icefall5 said:

Homework Statement


The matrix [itex]A = \begin {bmatrix} 0 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & 1 & 0 \end{bmatrix}[/itex] has two real eigenvalues, one of multiplicity 2 and one of multiplicity 1. Find the eigenvalues and a basis of each eigenspace.


Homework Equations


N/A


The Attempt at a Solution


I've done cofactor expansion to come up with the equation [itex] - \lambda^3 - \lambda^2[/itex],
That's not an equation. The equation is [itex] - \lambda^3 - \lambda^2[/itex] = 0, or ##\lambda^2(\lambda + 1) = 0##
Can you see that one of the eigenvalues has multiplicity 2 and the other has multiplicity 1?
icefall5 said:
and that the eigenvalues are therefore -1 and 0, but I don't know how to determine the multiplicity of each. I've looked it up and have gotten nowhere. I can do the rest of the problem, I just don't know how to get these multiplicities.

EDIT: Working on the problem further, and I got the basis for -1. There are supposed to be two vectors for 0, however, but the RREF of the matrix is [itex]\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}[/itex]. How does that translate to two eigenvectors?

Thanks in advance for any help!
 
  • #3
Ah, yes, I just couldn't simplify that properly; sorry! Thank you!

[STRIKE]But now the second part -- how does that matrix equate to two eigenvectors?[/STRIKE]

I'm an idiot; I figured it out. Thanks again!
 
Last edited:

1. What is the definition of multiplicity of eigenvalues?

The multiplicity of an eigenvalue of a square matrix is the number of times that eigenvalue appears as a root of the characteristic polynomial of the matrix.

2. How is multiplicity of eigenvalues related to eigenvectors?

The multiplicity of an eigenvalue is equal to the number of linearly independent eigenvectors associated with that eigenvalue. In other words, the number of eigenvectors that share the same eigenvalue is the multiplicity of that eigenvalue.

3. Can a matrix have eigenvalues with different multiplicities?

Yes, a matrix can have eigenvalues with different multiplicities. For example, a 2x2 matrix can have one eigenvalue with multiplicity 1 and another with multiplicity 2.

4. How does the multiplicity of eigenvalues affect the diagonalizability of a matrix?

If a matrix has distinct eigenvalues with multiplicities equal to 1, then it is diagonalizable. However, if a matrix has repeated eigenvalues, the matrix may still be diagonalizable depending on the number of linearly independent eigenvectors associated with each eigenvalue.

5. What if a matrix has a zero eigenvalue? What is its multiplicity?

If a matrix has a zero eigenvalue, its multiplicity is equal to the nullity of the matrix. This means that there are multiple linearly independent eigenvectors associated with the zero eigenvalue, and the matrix is not invertible.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
462
  • Calculus and Beyond Homework Help
Replies
2
Views
265
  • Calculus and Beyond Homework Help
Replies
10
Views
2K
  • Calculus and Beyond Homework Help
Replies
19
Views
3K
  • Calculus and Beyond Homework Help
Replies
6
Views
195
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
927
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
925
Back
Top