Hessian of a function

$ f $

The quadratic form

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

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)

Comments

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