Help understanding the geometry behind the maximum principle for eigenvalues. Context:
I am learning about the maximum principle, which states that

If $A$ is a real symmetric matrix and $q(x)=\langle Ax, x \rangle$, the following statements hold:

*

*$\lambda_1=\mathrm{max}_{||x||=1} q(x)=q(x_1)$ is the largest eigenvalue
of the matrix $A$ and $x_1$ is the eigenvector corresponding to
eigenvalue $\lambda_1$.

*Let $\lambda_k=\mathrm{max} q(x)$ subject to
the constraints   $\langle x,x_j \rangle=0 \ \ \ j=1,2,\dots ,k-1.$
$||x||=1$.


Then $\lambda_k=q(x_k)$ is the $kth$ eigenvalue of $A$, $\lambda_1\geq \lambda_2 \geq \dots \geq \lambda_k$ and $x_k$ is the corresponding eigenvector of $A$.
Now the book states that geometrically, we can visualize this as follows:

To understand the geometric significance of the maximum principle, we view the quadratic form $q(x)=\langle Ax,x \rangle$ as a radial map of the sphere. That is, to each point on the sphere $||x||=1$, associate $q(x)$ with the projection of $Ax$ onto $x$, $\langle Ax,x \rangle x$.If $A$ is positive definite, the surface $q(x)x$ is an ellipsoid. The maximum of $q(x)$ occurs exactly along the major axis of the ellipsoid. If we intersect it with the plane orthogonal to the major axis, then restricted to this plane, the maximum of q(x) occurs along the semi-major axis and so on.

I've bolded the parts that I don't understand. I don't see how this radial mapping is transforming the sphere into an ellipsoid nor the parts about the maximums occuring along the major axis and "so on". Could anyone elucidate this geometric idea for me?

The eigenvectors for a real symmetric matrix are a complete orthonormal set, that is we can take them a an orthonormal basis for $R^n$.So we can represent any $v$ on the unit $n$-sphere as $(v_j)_{1\le j \le n}$ in the sense that there is a unique $(v_j)_{1 \le j \le n}$ such that $v=\sum_j (v_j x_j)$, and furthermore we have $  \sum_j (v_j)^2=1$. That is, the lines from the origin ,through the eigenvectors, form a system of  mutually orthogonal co-ordinate axes. And $Av= \sum_j (\lambda_j v_j x_j).$
