# Eigenvalue and and eigenvector

1. Jun 5, 2014

### Gianfelici

Hi, I have a problem with the calculation of the eigenvalue of a matrix. That matrix is an N x N matrix which can be written as:

$M^{ab} = A\delta^{ab} + B \phi^a \phi^b$

where $\delta^{ab}$ is the identity matrix and the $\phi$ is a column vector. The paper I'm studying says that the eigenvalue of this matrix are:

A with molteplicity 1

$A + \phi^2 B$ with molteplicity N-1

but I can't understand why! Can anyone help me?

Last edited: Jun 5, 2014
2. Jun 5, 2014

You should put $around the latex code to render it 3. Jun 5, 2014 ### Gianfelici Thank you, now it'right 4. Jun 5, 2014 ### AlephZero I think it should be A with multiplicity N-1$A + \phi^2 B$with multiplicity 1. First find the eigenvalues of the rank 1 matrix$B\phi^a\phi^b$. Then think about what happens when you add$A\delta^{ab}##, which is A times the identity matrix.

5. Jun 11, 2014

### Gianfelici

yes, you're right, it was A with multeplicity N-1 and the other with multeplicity 1. I tried to use your suggestion and I solved it! thank you very much!