The Hessian Matrix and the Multivariable Second Derivative Test

The two-variable discriminant $D = f_{xx}f_{yy}-f_{xy}^2$ is the determinant of a $2\times 2$ matrix — the Hessian. In a portfolio of $n$ assets, the covariance matrix $\Sigma$ is itself the Hessian of the portfolio variance $\sigma^2_p = \mathbf{w}^\top\Sigma\mathbf{w}$; the condition for a minimum-variance portfolio to exist globally is that $\Sigma$ is positive definite. This note extends the discriminant test to $n$ variables: the Hessian $H$ of $f$ at a critical point is positive definite for a local minimum, negative definite for a local maximum, and indefinite for a saddle point. The tests for definiteness — eigenvalue signs and Sylvester's criterion (leading minor signs) — are the operative tools in any dimension.

Learning objectives. By the end of this note you should be able to: