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'' $  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|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|Pseudo-convex and pseudo-concave]]) and of a [[Plurisubharmonic function|plurisubharmonic function]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.M. Postnikov,   "Introduction to Morse theory" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR></table>

====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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.G. Krantz,   "Function theory of several complex variables" , Wiley  (1982)  pp. Chapt. 3</TD></TR></table>