Let $A$ be a symmetric $n×n$ matrix over $\mathbb{R}$, and let $q$ be the quadratic form on $\mathbb{R}^n$ given by $q(x_1,x_2,\cdots,x_n)=\sum_{i,j}A_{ij}x_ix_j$. Show that there is an invertible linear operator $U$ on $\mathbb{R}^n$ such that $q(U(x_1,x_2,\cdots,x_n))=\sum_{i=1}^n c_{i}x_i^2$, where $c_i$'s are $0,1$ or $-1$.
I have solved the above problem by using the fact that any symmetric bilinear form on a finite-dimensional real vector space can be represented by some diagonal matrix with diagonal entries in 0, 1 or -1 with respect to some ordered basis. But the next exercise 9 asks to prove the same fact using this problem. So, I was wondering if one can solve the problem without using the fact that symmetric bilinear forms are "diagonalizable" (in some sense).