A nice lemma one the eigenvalues of symmetric matrices

Question

$\newcommand{\ket}[1]{|#1\rangle}$ $\newcommand{\bra}[1]{\langle#1|}$ Let $M$ be an $n \times n$ real symmetric matrix, and $(e_1,...,e_n)$ and orthonormal basis of $\mathbb{R}^n$. Let $P_{ij} = \ket{e_i} \bra{e_i} + \ket{e_j} \bra{e_j}$ be the orthogonal projection onto $\text{span}(e_i,e_j)$, and we look at $P_{ij} M P_{ij}$ as $2 \times 2$ matrices (as their restriction to $\text{span}(e_i,e_j)$). The matrices $P_{ij} M P_{ij}$ are also symmetric. How to prove that \begin{align*} \forall i,j \in \{1, ...,n \}, P_{ij} M P_{ij} \text{ has two different eigenvalues } \\ \iff \text{ all the eigenvalues of $M$ are different} \end{align*}

Answer 1

This isn't true. Suppose $n=3,\,\{e_1,e_2,e_3\}$ is the standard basis of $\mathbb R^3$ and $$ M=\pmatrix{1&0&0\\ 0&\frac13&\frac{\sqrt{2}}{3}\\ 0&\frac{\sqrt{2}}{3}&\frac23}. $$ The eigenvalues of $M$ are $1,1$ and $0$, but each of $$ P_{12}MP_{12}\sim\pmatrix{1&0\\ 0&\frac13}, \ P_{13}MP_{13}\sim\pmatrix{1&0\\ 0&\frac23}, \ P_{23}MP_{23}\sim\pmatrix{\frac13&\frac{\sqrt{2}}{3}\\ \frac{\sqrt{2}}{3}&\frac23} $$ has two different eigenvalues (as none of them is a scalar matrix).