Let $M$ be a symmetric $n\times n$ tri-diagonal matrix, with positive values in its main diagonal. and let $\mathbf{1} \in R^n$ be the vector of all 1, such $M \mathbf{1} = 0$
Suppose $M$ has eigenvalues $0=\lambda_1 < \lambda_2 \leq \cdots \leq \lambda_n$, and let $\mathbf{v}_k$ be an eigenvector of $\lambda_{k}$, the corresponding $k$-th eigenvalue of $M$ with multiplicity 1.
Why $(M - \lambda_{k} I)$ has $k - 1$ negative eigenvalues?