3

Given two $n\times n$ symmetric real positive definite (p.d.) matrices $A$ and $B$, it is known that $$A \prec B \iff B^{-1} \prec A^{-1}$$ where $A\prec B$ means that $B-A$ is p.d..

Suppose $A\prec B$. If I know some information about the "magnitude" of $B-A$, e.g., the minimum eigenvalue of $B-A$ (which is positive), can I infer information about the "magnitude" of $A^{-1}-B^{-1}$, e.g., its minimum eigenvalue?

Rodrigo de Azevedo
  • 23,223
Uchiha
  • 923

1 Answers1

1

Consider $t\in(0,2/3)$ and $$B=\begin{bmatrix} 2&0\\0&1\end{bmatrix},\ \ \ A=\begin{bmatrix}1&0\\0&t\end{bmatrix}. $$ Then $$\lambda_\min(B-A)=1-t,\ \ \ \lambda_\min(A^{-1}-B^{-1})=\frac12.$$

Martin Argerami
  • 217,281