# Hessian of a function

$f$

$$H ( x) = \ \sum _ {i = 1 } ^ { n } \sum _ {j = 1 } ^ { n } a _ {ij} x _ {i} x _ {j} ,$$
$$H ( z) = \ \sum _ {i = 1 } ^ { n } \sum _ {j = 1 } ^ { n } a _ {ij} z _ {i} \overline{z}\; _ {j} ,$$
where $a _ {ij} = \partial ^ {2} f ( p)/ \partial x _ {i} \partial x _ {j}$( or $\partial ^ {2} f ( p)/ \partial z _ {i} \partial \overline{z}\; _ {j}$) and $f$ is given on the $n$- dimensional real space $\mathbf R ^ {n}$( or on the complex space $\mathbf C ^ {n}$) with coordinates $x _ {1} \dots x _ {n}$( or $z _ {1} \dots z _ {n}$). Introduced in 1844 by O. Hesse. With the aid of a local coordinate system this definition is transferred to functions defined on a real manifold of class $C ^ {2}$( or on a complex space), at critical points of the functions. In both cases the Hessian is a quadratic form given on the tangent space and is independent of the choice of coordinates. In Morse theory the Hessian is used to define the concepts of a (non-)degenerate critical point, the Morse form and the Bott form. In complex analysis the Hessian is used in the definition of a pseudo-convex space (cf. Pseudo-convex and pseudo-concave) and of a plurisubharmonic function.