Eigenvalues and Matrices

• August 30th 2010, 11:39 AM
Ian1779
Eigenvalues and Matrices
Hi

Just after a bit of pointing in the right direction. I have 3 eigenvalues and I am trying to find the matrix that they correspond to.

How do I do it? It is really stumping me!!

Thanks very much
• August 30th 2010, 12:05 PM
Ackbeet
Are you trying to find the eigenvectors? Or are you trying to find the original matrix $A$ from its eigenvalues?
• August 30th 2010, 12:12 PM
Ian1779
The original matrix $A$
• August 30th 2010, 12:15 PM
Ackbeet
Well, I'm afraid you're not going to be able to determine the matrix A uniquely. There are, in general, infinitely many matrices that are similar to A that have the same eigenvalues. Do you know the size of A?
• August 30th 2010, 12:29 PM
Ian1779
I believe the matrix size is 3x3
• August 30th 2010, 12:32 PM
Ackbeet
In that case, assuming you have three distinct eigenvalues, the best you can do is construct a diagonal matrix similar to A. That is, you can construct a diagonal matrix $D$ such that there exists an invertible matrix $P$ such that $A=PDP^{-1}.$ You construct $D$ by simply having your 3 eigenvalues on the main diagonal, and zeros everywhere else.

If you have more conditions on A, you might be able to say more than this.
• August 30th 2010, 12:36 PM
Ian1779
I will give that a go and see how I get on - Cheers
• August 30th 2010, 12:52 PM
Ackbeet
Right-ho.
• August 30th 2010, 01:07 PM
Ian1779
It worked - I got the answer I should have - thanks again!!
• August 30th 2010, 01:08 PM
Ackbeet
You're very welcome. Have a good one!