I'm a beginner of matrix inequality. Now I have a vector inequality like below
\begin{equation} \begin{aligned} A \lambda \succeq \mu \end{aligned} \end{equation}
where $A$ is a $n \times n$ symmetric positive definite matrix and $\lambda \succeq 0$ is a $n \times 1$ vector. May I ask if I can draw the following conclusion?
\begin{equation} \begin{aligned} \lambda \succeq A^{-1} \mu \end{aligned} \end{equation}
If the equation above is correct, how can I prove it?
Thanks in advance!
No, you cannot. Choose $\lambda=0$; certainly $0\succeq\mu$ does not imply $0\succeq A^{-1}\mu$, for instance for $\mu=\pmatrix{-1\\0}$ and $A^{-1}=\pmatrix{3&-1\\-1&3}$.