Hessian of a function

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The quadratic form


where (or ) and is given on the -dimensional real space (or on the complex space ) with coordinates (or ). 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 (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.


[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 on 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.


[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:
This article was adapted from an original article by L.D. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article