# The reverse Cauchy-Schwarz inequality

Let  $I$ be a symmetric, hyperbolic bilinear form (that is, of signature $+,+,\dots,+,-$) over a finite dimensional real vector space (though this may hold in general as well). If $x$ is positive, meaning $I(x,x)>0$, then for all $y$ we have

$I(x,y)^2 \geq I(x,x)I(y,y).$

1. One is reduced to show that for vectors $x,y$ and scalars $a,b$ subject to $|x|^2 >a^2$, we have $( |x|^2-a^2)(|y|^2- b^2)\leqslant (\langle x,y\rangle -ab)^2$. One can rewrite the above as a quadratic polynomial in $b$, which has discriminant precisely $(|x|^2-\alpha^2)(\langle x,y\rangle^2-|x|^2|y|^2)$, whence the result.