Let $A, B$ be $n \times n$ positive definite matrices, and let $x$ be an $n$-vector. Why does the following hold?
$$ x^{T}(A^{-1/2} (A^{1/2} B^{-1} A^{1/2})^{1/2} A^{-1/2})x \leq (x^{T}A^{-1}x)^{1/2}(x^{T}B^{-1}x)^{1/2} $$
2026-03-26 06:33:44.1774506824
Why does this inequality with quadratic forms hold?
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Presumably $x$ is real. Let $u=A^{-1/2}x$ and $v=(A^{1/2}B^{-1}A^{1/2})^{1/2}A^{-1/2}x$. Then the inequality in question is just Cauchy-Schwarz inequality $\langle u,v\rangle\le\|u\|_2\|v\|_2$ in disguise.