Schur Complement - Schur Complement Condition For Positive Definiteness

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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