How to determine which eigenvalue has multiplicity 2 ?

In summary, to determine which eigenvalue has a multiplicity of 2 in a 4x4 matrix without calculating the characteristic polynomial or determining the dimensions of (A-λI), you can use the definition of trace to set up an equation and solve for the remaining eigenvalue. This method is quicker than row reducing multiple matrices and can be easily applied during an exam.
  • #1
sid9221
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Say I have a 4x4 matrix and I know 3 eigenvalues and the 3 corresponding eigenvectors.

Is there a fast way to calculate which one has multiplicity 2 without calculating the characteristic polynomial(too time consuming for a 4x4 matrix) or without determining the dimensions of (A-λ I) for each eigenvalue λ. (Row reducing 3 4x4 matrices is too long during an exam)
 
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  • #2
LOL i just though of the solution as soon as I typed the question.

I guess I could set up an equation using the definition of trace:

Ʃ(elements of diagnal)=λ1+λ2+λ3+x

hence work out x, which will tell me which eigenvalue is repeated.
 
  • #3
nice! :smile:
 

1. What is an eigenvalue with multiplicity 2?

An eigenvalue with multiplicity 2 means that it has a geometric multiplicity of 2, which indicates that there are two linearly independent eigenvectors associated with that eigenvalue.

2. How do I determine if an eigenvalue has multiplicity 2?

To determine if an eigenvalue has multiplicity 2, you can use the characteristic equation of the matrix and solve for the roots. If a particular eigenvalue appears twice as a root, it has multiplicity 2.

3. What does it mean if an eigenvalue has multiplicity 2?

If an eigenvalue has multiplicity 2, it means that there are two linearly independent eigenvectors associated with that eigenvalue. This can occur when there is a repeated root in the characteristic equation or when the matrix has a repeated eigenvalue.

4. Can an eigenvalue have multiplicity greater than 2?

Yes, an eigenvalue can have multiplicity greater than 2. This means that there are more than two linearly independent eigenvectors associated with that eigenvalue.

5. How do I find the eigenvectors for an eigenvalue with multiplicity 2?

To find the eigenvectors for an eigenvalue with multiplicity 2, you can use the eigenspace method. This involves finding the null space of the matrix (A-λI)^2, where λ is the eigenvalue with multiplicity 2. The basis vectors of this null space will be the eigenvectors associated with that eigenvalue.

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