Finding the Basis for Repeated Eigenvalues: Explained

[mathrocks]

I'm trying to find the basis for a particular matrix and I get a 3 eigenvalues with two of them being identical to each other. What do I do to find the basis for the repeated eigenvalue? Will it have the same basis as the original number?

Thanks!

 

[matt grime]

The eigenvectors may or may not span the original space. That is there may only be two eigenvectors. If you solve as usual for the eigenvector you may obtain two linearly independent vectors for it (the repeated eigenvalue) or you may only get one.

If you want the geometric interpetation of this then you need to learn about Jordan Normal Form, or Jordan Canonical Form.

 

[M98Ranger]

When dealing with repeated eigenvalues, it is important to remember that each eigenvalue corresponds to a unique eigenvector. This means that even though two eigenvalues may be identical, their corresponding eigenvectors may be different. So, to find the basis for the repeated eigenvalue, you will need to find all the linearly independent eigenvectors associated with that eigenvalue.

To do this, you can use the method of elimination. Start by finding one eigenvector for the repeated eigenvalue by solving the characteristic equation (det(A-λI)=0) and plugging in the repeated eigenvalue. Then, find a second eigenvector by plugging in the same eigenvalue but using a different basis vector. Continue this process until you have found all the linearly independent eigenvectors associated with the repeated eigenvalue.

It is also important to note that the basis for the repeated eigenvalue may not be the same as the original basis. This is because the eigenvectors associated with the repeated eigenvalue may be different from the original eigenvectors. However, the basis for the repeated eigenvalue will still span the same subspace as the original basis.

In summary, when dealing with repeated eigenvalues, you will need to find all the linearly independent eigenvectors associated with that eigenvalue to determine the basis. This basis may be different from the original basis, but it will still span the same subspace. I hope this helps clarify the process for finding the basis for repeated eigenvalues.