$(x^*Ay)^2 \leq (x^*Ax)(y^*Ay)$
Does this inequality hold for positive semi-definite matrix A? I wonder if this is equivalent to Cauchy-Schwarz Inequality. I tried to diagonalize A but it still half-way towards Cauchy-Schwarz Inequality.
Any thoughts on this inequality? Am I going on the right direction?
Thanks a lot!
Best Answer
The Cauchy-Schwarz inequality applies to any semi-definite inner product, including the one on $\mathbb{R}^n$ (as column vectors) defined by $\langle x,y\rangle=x^{\mathrm T}Ay$.
One of the simplest proofs in the real case is to consider $p(t)=\langle x-ty,x-ty\rangle=\langle x,x\rangle -2\langle x,y\rangle t +\langle y,y\rangle t^2$. Then $p(t)\geq 0$ for all $t\in \mathbb R$, and $p$ is a quadratic polynomial in $t$, so its discriminant must be nonpositive. Rearranging this discriminant inequality yields your inequality.