Problem on Minkowski's reduction theory of positive definite matrix

Question

I am facing problem to understand Minkowski's reduction theory from Klingen's Siegel Modular form book. I am giving the pictures of the corresponding pages. I am stuck there for hours. $P_n=\{y\in M(n,\Bbb R)| y>0\}$

Now my question is

1. "What does minimal mean where he states "$y[u_1]$ becomes minimal"?
2. If the minimal means minimum then how to prove that $\exists u_1 \neq 0$ s.t $y[u_1]=u_1^tyu_1\neq 0$?
3. How to show there are finitely many integral $u_1$ which are primitive(Definition of primitive is given later on)?
4. Then what is the guarantee of the existence of such $u_k$?
5. Why can we replace $y_k$ by $-y_k$?
6. And explain the later condition as well.

In short, if some can elaborate this lines or explain me with an example then also it would be fruitful as I can't proceed by just skipping it.

I am sorry for asking questions in this way but I can't find some other way out. Please help.

Answer 1

Minkowski reduction is usually thought of in terms of quadratic forms (positive). There is no real difference discussing for matrices, symmetric real and positive definite. I was able to read pages 1-10 of Klingen, that is exactly what you have. You also should consider buying the inexpensive Cassels, Rational Quadratic Forms, he leads directly from reduction to Siegel Domains.