|
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. 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". Specifically, suppose is a function taking as input a vector and outputting a scalar ; if all second partial derivatives of exist and are continuous over the domain of the function, then the Hessian matrix of is a square matrix, usually defined and arranged as follows: : or, component-wise: : The determinant of the above matrix is also sometimes referred to as the Hessian. The Hessian matrix can be considered related to the Jacobian matrix by . ==Mixed derivatives and symmetry of the Hessian== The mixed derivatives of ''f'' are the entries off the main diagonal in the Hessian. Assuming that they are continuous, the order of differentiation does not matter (Clairaut's theorem). For example, : In a formal statement: if the second derivatives of are all continuous in a neighborhood , then the Hessian of is a symmetric matrix throughout ; see symmetry of second derivatives. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hessian matrix」の詳細全文を読む スポンサード リンク
|