I'm trying to do Exercise 3.6 at the end of this pdf (in $\Bbb CP^2$):
- Show that the two quadratic forms $$x^2+y^2-z^2, \quad x^2+y^2-y z$$ cannot be simultaneously diagonalized.
Attempt 1:
Their matrices don't commute: $$\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right) \cdot\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & -\frac{1}{2} \\ 0 & -\frac{1}{2} & 0\end{array}\right)\ne\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & -\frac{1}{2} \\ 0 & -\frac{1}{2} & 0\end{array}\right) \cdot\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right)$$ but simultaneously diagonalizable matrices commute, so they cannot be simultaneously diagonalized.
Then I realize I am wrong. To diagonalize quadratic forms we want $S^TAS$ and $S^TBS$ to be diagonal matrices, but in the linked is another question: $S^{-1}AS$ and $S^{-1}BS$ are diagonal matrices.
Attempt 2:
I try to relate this exercise to a proposition in the PDF:
The non-diagonalizable case is where the intersection is at an odd number of points:
The two conics $$x^2+y^2-z^2=0, \quad x^2+y^2-y z=0$$ intersect at two points $[1,\pm i,0]$ with multiplicity $1$, and one point $[0,1,1]$ with multiplicity $2$. So the intersection is at an odd number of points, so they cannot be simultaneously diagonalized.
But how do we prove this proposition?
Attempt 3:
For the first quadratic form $x^2+y^2-z^2$, let $z'=iz$, then $x^2+y^2-z^2=x^2+y^2+z'^2$, then the second quadratic form $x^2+y^2-yz=x^2+y^2+iyz'$, so it has matrix $\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & \frac{i}{2} \\ 0 & \frac{i}{2} & 0\end{array}\right)$.
The Jordan normal form of $\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & \frac{i}{2} \\ 0 & \frac{i}{2} & 0\end{array}\right)$ is $J=\left(\begin{array}{lll}\frac{1}{2} & 1 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & 1\end{array}\right)$ WA computation
this is non-diagonalizable over $\Bbb C$, since it has a Jordan block of size $2$.
Is it true in general that, if $J$ is a $3\times3$ matrix with a Jordan block of size $>1$, and $I_3$ is the identity matrix, then the two corresponding conics has odd number of intersections?
