I have a list of vectors $v_1, \cdots, v_m \in \mathbb{R}^n$ with $v_i \not=0$ and i'm trying to better understand the eigenspaces of the matrix $M = v_1 {v_1}^T + \cdots + v_m v_m^T$, which is diagonalizable since it's symmetric. We can assume $m<n$. There are some pretty easy things to notice about it. For example $E(0, M)$, the eigenspace with eigenvalue $0$, is the set of vectors in $\mathbb{R}^n$ orthogonal to all $v_i$. Also, if $\lambda\not=0$ then $E(\lambda, M) \subseteq Span(v_1, \cdots, v_m)$.
But I want to understand more deeply the connection between the vectors $v_i$ and the eigenspaces of $M$. For example:
- What are the conditions on the vectors $v_i$ for $M$ to have eigenvalues $\lambda \not=0$ with multiplicity $>1$?
- Is there a "simpler" way to compute the eigenvalues of $M$ directly from the $v_i$ without having to find the characteristic polynomial of $M$?