Eigenvalues of nxn matrix with variable coeficients

Question

I would like help for find the maximum and minimum eingenvalues of the following symmetric matrix or order $n$ , $A=I_{n}+(p(x)-2)(\xi \otimes \xi )|\xi|^{-2}$.

Where

• $I_{n}$ is the identity matrix of order $n$

• $\xi \in \mathbb{R}^{n}$

• $p \in C^{1}(B_{1})$

• $\xi \otimes \xi$ is the matrix where the $(i,j)$ element is $(\xi_{i}\xi_{j})$

• $B_{1}$ is unit ball in $\mathbb{R}^{n}$

My conjecture is that the minimum eingenvalue is $\min(p(x)-1,1)$ ans the maximum eingenvalue is $\max(p(x)-1,1)$, because in the case $n=2$ this is true, but i don't know if the dimension obstructs this fact.

Any help?

Thanks

Answer 1

Yup.

$\xi$ is an eigenvector with eigenvalues $p(x) - 1$.

All vectors orthogonal to $\xi$ are eigenvectors with eigenvalue $1$.