Inequality for Quadratic Forms

Question

Suppose $\boldsymbol{c}>0$ is a vector of 2 elements. Suppose $A$ and $B$ are $2 \times 2$ pd matrices such that $\boldsymbol{c}'A\boldsymbol{c} \le \boldsymbol{c}'B \boldsymbol{c}$. Is $\boldsymbol{c}'A^{-1}\boldsymbol{c} \ge \boldsymbol{c}'B^{-1} \boldsymbol{c}$?

Answer 1

Hint:

• If $A-B$ is positive definite, then $B^{-1}-A^{-1}$ is positive definite.
• If $A-B$ is positive definite, then $c^{T}Ac>c^{T}Bc~,~\forall c$