Let $Y$ be diaogalizable matrix $Y = PDP^\top$, where $P$ is orthogonal and $X$ be congruent to $Y$, that is there exists an invertible matrix $A$ such that $X=A^\top YA$. Thus we can find a linearly independent basis in which $X$ is a diagonal matrix, namely $X = (P^\top A)^\top D(P^\top A)$. Now if I want to find a lower bound estimate of the minimum eigen-value of $X$ in terms of $D$ how can I proceed?
My solution: $$\sigma_\min(X) = \frac{1}{\|X^{-1}\|}$$ $$\|X^{-1}\| \leq \|P\|^2\|A^{-1}\|^2\|D^{-1}\|=\|A^{-1}\|^2\|D^{-1}\|$$ Therefore, $\sigma_\min(X) =\frac{1}{\|X^{-1}\|} \geq \sigma_\min(A)^2\sigma_\min(D)$