how to find if the determine is a saddle point, minima, or maxima?

Question

i have a 3 by 3 hessian matrix, how do i ssolve for the determinant and identify if it is a saddle point , minima or maxima

Answer 1

Given a Hessian about a critical point, you want the signs of the eigenvalues to determine whether it's a minimum, maximum, or a saddle. If the eigenvalues are all real and positive, the critical point is a local minimum; that is, moving in any basis direction results in an increase. If the eigenvalues are all real and negative, the critical point is a maximum. Lastly, if the eigenvalues are of mixed sign, the point is a saddle.

In two variables, you have a 2x2 Hessian, and the determinant actually tells you whether the signs agree, since the determinant is the product of the eigenvalues. That is, if the determinant's negative in the 2x2 case, you know one eigenvalue is negative and one is positive, and so the point must be a saddle. In the 3x3 case it's not as straightforward; different combinations of signs can give both positive and negative determinants. +++ and --+ both give positive determinant; -++ and --- both give negative determinant. So in the 3x3 case you need to compute the eigenvalues directly to find the result.

Answer 2