Sylvester's Criterion
A real symmetric matrix of size is positive definite iff its signature is , and negative definite iff its signature is .
Proposition 9.8 (Sylvester's Criterion)
Let be a real symmetric matrix of size . Then
Application to stationary points
For the stationary points of , the Hessian matrix is .
At the leading principal minors give and : the signature matches neither admissible pattern, and since all eigenvalues are non-zero, so the Hessian is indefinite and is a saddle point.
At the minors give and . The signature matches the positive definite pattern, and hence is a local minimum.
Related
Stated in
- Proposition 9.8 (Sylvester's Criterion)§9.3.2 Nature of Stationary Points and the Hessian