Why Are the Eigenvalues of This Matrix A and A + φ²B?

[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?

 

[adjacent]

You should put ## around the latex code to render it

 

[Gianfelici]

You should put ## around the latex code to render it

Thank you, now it'right

 

[AlephZero]

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

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.

 

[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!