# Hessian of a function

(Redirected from Hessian matrix)

$f$

$$H ( x) = \ \sum _ {i = 1 } ^ { n } \sum _ {j = 1 } ^ { n } a _ {ij} x _ {i} x _ {j} ,$$

or

$$H ( z) = \ \sum _ {i = 1 } ^ { n } \sum _ {j = 1 } ^ { n } a _ {ij} z _ {i} \overline{z} _ {j} ,$$

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.

#### References

 [1] M.M. Postnikov, "Introduction to Morse theory" , Moscow (1971) (In Russian) [2] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)

One usually calls the form $H ( z)$ on $\mathbf C ^ {n}$ the complex Hessian.