# Why can a constraint on a matrix being positive definite be rewritten as the matrix minus the identity being positive semidefinite?

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?

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