Schur Complement Condition For Positive Definiteness
Let X be a symmetric matrix given by
Let S be the Schur complement of A in X, that is:
Then
- is positive definite if and only if and are both positive definite:
- .
- is positive definite if and only if and are both positive definite:
- .
- If is positive definite, then is positive semidefinite if and only if is positive semidefinite:
- , .
- If is positive definite, then is positive semidefinite if and only if is positive semidefinite:
- , .
These statements can be derived by considering the minimizer of the quantity
as a function of u (for fixed v).
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