| # Hessian of a Function ## Definition The Hessian of a twice-differentiable function at a point is the matrix containing the second derivatives of the function at that point. That is, the Hessian is the matrix with elements given by The Hessian of at is often denoted . The second-derivative is independent of the order in which derivatives are taken. Hence, for every pair . Thus, the Hessian is a symmetric matrix. ## Examples ### Hessian of a quadratic function Consider the quadratic function The Hessian of at is given by For quadratic functions, the Hessian is is a constant matrix, that is, it does not depend on the point at which it is evaluated. ### Hessian of the log-sum-exp function Consider the ‘‘log-sum-exp’’ function , with values The gradient of at is where , . The Hessian is given by More generally, the Hessian of the function with values is as follows. First the gradient at a point is (see here): where , and . Now the Hessian at a point is obtained by taking derivatives of each component of the gradient. If is the -th component, that is, then and, for : More compactly: