In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a function. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants".
Given the real-valued function
if all second partial derivatives of f exist and are continuous over the domain of the function, then the Hessian matrix of f is
where x = (x1, x2, ..., xn) and Di is the differentiation operator with respect to the ith argument. Thus
Because f is often clear from context, is frequently abbreviated to .
The Hessian matrix is related to the Jacobian matrix by = .
Hessian matrices are used in large-scale optimization problems within Newton-type methods because they are the coefficient of the quadratic term of a local Taylor expansion of a function. That is,
where J is the Jacobian matrix, which is a vector (the gradient) for scalar-valued functions. The full Hessian matrix can be difficult to compute in practice; in such situations, quasi-Newton algorithms have been developed that use approximations to the Hessian. The best-known quasi-Newton algorithm is the BFGS algorithm
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Specifically, suppose is a function (which takes as input real n-tuples and produces as output real m-tuples). Such a function is given by m real-valued component functions, . The partial derivatives of all these functions with respect to the variables (if they exist) can be organized in an m-by-n matrix, the Jacobian matrix of , as follows: