I know we can use orthogonal diagonalization to identify the conic $$5x^2 − 4xy + 8y^2 − 36 = 0$$ by rotating the $xy$-axes and put it in standard position.
But why do we need a rotation? What happens if we use a reflection instead?
Update:
For example, consider $x^2 + 2xy+y^2 -3x-5y+4=0$. The matrix $$ P = \frac{1}{\sqrt{2}} \left( \begin{matrix} 1 & 1 \\ -1 & 1 \end{matrix} \right) $$ is a rotation and with it I get the expression: $\sqrt{2}s+ 2t^2=0.$ A parabola.
But if I use $$ Q = \frac{1}{\sqrt{2}} \left( \begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix} \right) $$ a reflection, then I get $2s^2+\sqrt{2}t=0.$ Again, a parabola.