Linear Algebra and Its Applications · Problem 6.2.9

The key is recognizing that $A = R^T R$ lets you rewrite the quadratic forms as inner products of transformed vectors. If you set $u = Rx$ and $v = Ry$, then $x^T A x = u^T u$ and $y^T A y = v^T v$—turning the inequality into the standard Cauchy–Schwarz inequality $|u^T v|^2 \leq (u^T u)(v^T v)$, which you already know is true. The factorization essentially "linearizes" the problem.

The key is recognizing that $A = R^T R$ lets you rewrite the quadratic forms as inner products of transformed vectors. If you set $u = Rx$ and $v = Ry$, then $x^T A x = u^T u$ and $y^T A y = v^T v$—turning the inequality into the standard Cauchy–Schwarz inequality $|u^T v|^2 \leq (u^T u)(v^T v)$, which you already know is true. The factorization essentially "linearizes" the problem.