Finding diagonal transformation matrix of a bilinear form

Question

Let $f:\mathbb R^3 \times \mathbb R^3 \rightarrow \mathbb R$ be a symmetric bilinear form, and let $q$ be its quadric form, so that $q(x, y, z)= xy+yz$.

1. Find the transformation matrix $A$ of $f$ by the standard basis.

2. Find invertiable matrix $Q$ and diagonal matrix $D$ so that $A=Q^tDQ$

My solution for 1 is: $\begin{matrix} 0& 1\over2& 0\\ 1\over2&0&1\over2\\0&1\over2&0\end{matrix}$

I didn't manage solving 2. I used matrix congruence and solving for the block matrix [A|I], but I just didn't manage bringing A to a diagnoal form, because the columme operations always cancelled out the row operations.

Thanks in advance!

Answer 1

Hint

Every symmetric real matrix is diagonalizable by an orthogonal matrix.