The $n \times n$ real symmetric matrix $M$ has eigenvectors of unit length $\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}$, with corresponding eigenvalues $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$, where $\lambda_{1}>\lambda_{2}>\cdots>\lambda_{n}$. Prove that the eigenvalues are real and that $\mathbf{e}_{a} \cdot \mathbf{e}_{b}=\delta_{a b}$.

Let $\mathbf{x}$ be any (real) unit vector. Show that

$$\mathbf{x}^{\mathrm{T}} M \mathrm{x} \leqslant \lambda_{1}$$

What can be said about $\mathbf{x}$ if $\mathbf{x}^{\mathrm{T}} M \mathbf{x}=\lambda_{1} ?$

Let $S$ be the set of all (real) unit vectors of the form

$$\mathbf{x}=\left(0, x_{2}, \ldots, x_{n}\right)$$

Show that $\alpha_{1} \mathbf{e}_{1}+\alpha_{2} \mathbf{e}_{2} \in S$ for some $\alpha_{1}, \alpha_{2} \in \mathbb{R}$. Deduce that

$$\underset{\mathbf{x} \in S}{\operatorname{Max}} \mathbf{x}^{\mathrm{T}} M \mathbf{x} \geqslant \lambda_{2}$$

The $(n-1) \times(n-1)$ matrix $A$ is obtained by removing the first row and the first column of $M$. Let $\mu$ be the greatest eigenvalue of $A$. Show that

$$\lambda_{1} \geqslant \mu \geqslant \lambda_{2}$$