Hessians

A Hessian is a mathematical tool used to analyze how a function changes in multiple directions. Specifically, it’s a matrix that captures all the second-order partial derivatives of a function, indicating how its slope or rate of change itself varies. In practical terms, the Hessian helps determine whether a point is a local minimum, maximum, or saddle point, by providing information about the curvature of the surface defined by the function. It’s a key concept in optimization and advanced calculus, offering insight into the shape and behavior of complex systems.