= Simultaneous diagonalization of two transverse conics
{title2=$P^TAP,\ P^TBP\text{ diagonal}$}
Two <complex plane> <conics> meeting in four distinct points can be diagonalized by the same invertible change of homogeneous coordinates. Their intersection points form a <projective frame>: three on a line would force that line to be a common component by <Bézout theorem>, contradicting a finite intersection. Move the four points to the coordinate points and $[1:1:1]$. Each equation becomes $ayz+bzx+cxy=0$ with $a+b+c=0$. Substituting $x=v+w,y=w+u,z=u+v$ gives $au^2+bv^2+cw^2$; the substitution matrix has determinant two. Composing this with the frame change yields the displayed simultaneous congruence of the symmetric coefficient matrices. This is congruence of quadratic forms, not a claim of simultaneous similarity of arbitrary matrices.
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