I'm currently learning about local extrema in serveral variables and have come across the second derivative test for classifying critical points of multivariable functions.
I have read and understood the test (see link below), however I don't understand the idea behind it. Why is the critical point of a function a minimum if the eigenvalues of the Hessian matrix are all positive? I understand the idea behind the single variable case, however I am confused about the role of eigenvalues in the case of several variables.
Any insight into this would be much appreciated.