# Hessian of a function

$f$

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

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.

If the Hessian of a real-valued function is a positive (semi-) definite form, then the function is convex; similarly, if the complex Hessian of a function is a positive (semi-) definite form, then the function is plurisubharmonic.

#### References

 [a1] S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Chapt. 3
How to Cite This Entry:
Hessian of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hessian_of_a_function&oldid=47222
This article was adapted from an original article by L.D. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article