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Let $A$ be symmetric positive definite matrix and $E$ is symmetric with $||E||_{2} < ||A^{-1}||^{-1}_{2}$ then prove that $A+E$ is symmetric positive definite.

-- \ Observation; Since $A$ is invertible and $A+E = A(I+A^{-1}E)$ since $||A^{-1}E||_{2} \leq ||A^{-1}||_{2}||E||_{2}< 1$ by assumption then $A+E$ is invertible. But I don't see the connection to show that eigenvalues of $A+E$ are positive.

Any hint would be appreciated.

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    You could replace $E$ by $tE$ and your argument shows that $A+tE$ is invertible for all $0\leq t\leq1$. As $t$ goes from $0$ to $1$, I think you could show that eigenvalues must all remain positive by continuity.2011-12-08

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