Principal axis transformation of a symmetric matrix | Cornelius Lanczos

Question

I was reading the book Linear Differential Operators by Cornelius Lanczos Chapter3:page106 . I need help understanding section 3.4 about the Principal axis transformation of a symmetric matrix. The screenshot of the section is attached herewith,

1. I do not know what a second-order surface in an n-dimensional space means.
2. How eqn:3.4.6 and 3.4.7 are said to be satisfying each other.
3. A matrix example in numerical will be a great help for me to understand this abstract.

Answer 1

1. Also called quadric, here an ellipsoid. A surface defined as the solution of a second degree equation. There is some fluidity between "degree" and "order", often in the context of polynomials or power series the order is the lowest degree of an occurring term. For instance in truncating a power series at degree $d$, the order of the remainder is $d+1$ (or higher).

2. By definition of the surface and the eigenvalue equation $1=u^TSu=u^T(Su)=u^T(\lambda u)=λ(u^Tu)$.