What is the condition for the saddle point of a function of three variables?

Question

For a function $f(x,y)$ of two real variables $x$ and $y$, a point $(x_0,y_0)$ is a saddle point if the determinant of the Hessian matrix $$[f_{xx}f{yy}-(f_{xy})^2]_{x=x_0,y=y_0}<0.$$ If we are given a function $f(x,y,z)$ of three real variables $x,y$ and $z$, is there a similar criterion for a point $(x_0,y_0,z_0)$ to be a saddle point in terms of the determinant of Hessian matrix?

Answer 1

When we have $3$ (or, in general, $n$) variables, a point $(x_0,y_0,z_0)$ is a saddle point if the gradient is null at that point and if the Hessian has both positive and negative eigenvalues. Since the determinant is the product of all eigenvalues, in dimension $3$ its sign could be both positive (two negative eigenvalues, one positive one) or negative (two positive eigenvalues, one negative one) on a saddle point. Thus the answer to your question is no, the determinant of the Hessian does not give enough information to characterize a saddle point if we have $3$ or more variables.