Let $A$ be an invertible matrix with positive eigenvalues and $B$ be a positive definite matrix . How to estimate the minimum eigenvalues of $AB$ by using the eigenvalues of $A$ and $B$. More precisely, are there some inequalities such as $\min\{\lambda(AB)\}\geq\min\{\lambda(A),\lambda(B)\}$? Here $\lambda(A)$ denotes the eigenvalue of $A$. Thank you.


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