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

You look at the eigenvalues. If all three are positive, you have a relative minimum, if all three are negative, you have a relative maximum, and if you have both positive and negative values, you have a saddle point. If you have all 0s and positives or all 0s and negatives, further investigation is required.

You solve for eigenvalues by subtracting x from the diagonal elements, calculating the determinant, and finding the x values and setting it equal to 00. Alternatively, if you can find another way to show whether the eigenvalues are 0, positive, or negative, that works, too.