1) How do you manage to transform a matrix from quadratic to canonical form?
For instance, assume a linear transformation such that: $$Q(x,y,z)=x^2+2xz+z^2;$$
As far as I can see, in the canonical form this would just be $a^2+b^2$, $a=x+z$ and $b=y$, but the correct answer is something entirely different according to my book.
2) Assume for instance the linear transformation $Q(x,y,z)=x^2+xy+xz+yz$. How would you find its quadratic matrix $Q$, its symmetric part $B$ (I assume it is the sum of the quadratic and the quadratic transposed divided by $2$, but I'm not entirely sure), a diagonal quadratic form $G$ and the diagonalizing matrix $S$?
Notice that there isn't unicity of decomposition.
We have for your example \begin{align}Q(x,y,z)&=x^2+xy+xz+yz=x^2+x(y+z)+yz\\&=\left(x+\frac{1}{2}(y+z)\right)^2-\frac{1}{4}(y+z)^2+yz\\&=\left(x+\frac{1}{2}(y+z)\right)^2-\frac{1}{4}y^2-\frac{1}{4}z^2+\frac{1}{2}yz\\&=\left(x+\frac{1}{2}(y+z)\right)^2-\frac{1}{4}(y-z)^2\end{align}
The matrix of the quadratic form is $$A=\left(\begin{matrix}1&\frac{1}{2}&\frac{1}{2}\\ \frac{1}{2}&0&\frac{1}{2}\\ \frac{1}{2}&\frac{1}{2}&0\end{matrix}\right)$$ and we have $$Q(x,y,z)=(x,y,z)A \left(\begin{matrix}x\\ y\\ z\end{matrix}\right)$$