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My instructor today mentioned that if we have a constraint that a matrix $A$ is positive definite, then we can rewrite this constraint as $A - I$ is positive semidefinite without this affecting the feasibility of the problem.

Why wouldn't this affect the feasibility? The eigenvalues of $A - I$ would be one less than all the eigenvalues of A, so if A has an eigenvalue = $1/2$, wouldn't $A - I$ have an eigenvalue that is $-1/2$, changing the feasibility of the problem?

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    $\begingroup$ should be $A- \varepsilon I$ $\endgroup$ – Will Jagy Jan 17 at 21:21

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