What is a Hessian matrix?

I know that the Hessian matrix is a kind of second derivative test of functions involving more than one independent variable. How does one find the maximum or minimum of a function involving more than one variable? Is it found using the eigenvalues of the Hessian matrix or its principal minors?

You should have a look here: https://en.wikipedia.org/wiki/Second_partial_derivative_test

For an n-dimensional function f, find an x where the gradient grad f = 0. This is a critical point.

Then, the 2nd derivatives tell, whether x marks a local minimum, a maximum or a saddle point.

The Hessian H is the matrix of all combinations of 2nd derivatives of f.

1. For the 2D-case the determinant and the minors of the Hessian are relevant.
2. For the nD-case it might involve a computation of eigen values of the Hessian H (if H is invertible) as part of checking H for being positive (or negative) definite.

In fact, the shortcut in 1) is generalized by 2)

For numeric calculations, some kind of optimization strategy can be used for finding x where grad f = 0.