I know what is a Hessian matrix, and mathematical calculations required to calculate if a matrix is positive-semidefinite.
Can someone explain geometrically, why do we check if the Hessian matrix of a multivariable function is positive semidefinite, to determine if the function is convex?
Best Answer
As in calculus, if a function has nonnegative second derivative in its domain, then it is a convex function. For multivariable functions, the hessian to be positive semidefinite is telling you exactly the same information. The geometric intuition of being positive semidefinite at one point is that the graph of that function is above the tangent plane.
Positive definiteness says that all the eigenvalues are positive, which means that any time you look along an eigenvector then the function is curving up. If the Hessian is nondegenerate, then the eigenvectors form a basis near that point and in any direction, you'll also see "curving up" because you can decompose the direction into eigenvector directions.