Difference between revisions of "Hessian of a function"
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H ( z) = \ | H ( z) = \ | ||
\sum _ {i = 1 } ^ { n } \sum _ {j = 1 } ^ { n } | \sum _ {i = 1 } ^ { n } \sum _ {j = 1 } ^ { n } | ||
| − | a _ {ij} z _ {i} \overline{z} | + | a _ {ij} z _ {i} \overline{z} _ {j} , |
$$ | $$ | ||
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or $ \partial ^ {2} f ( p)/ \partial z _ {i} \partial \overline{z}\; _ {j} $) | or $ \partial ^ {2} f ( p)/ \partial z _ {i} \partial \overline{z}\; _ {j} $) | ||
and $ f $ | and $ f $ | ||
| − | is given on the $ n $- | + | is given on the $ n $-dimensional real space $ \mathbf R ^ {n} $ (or on the complex space $ \mathbf C ^ {n} $) |
| − | dimensional real space $ \mathbf R ^ {n} $( | + | with coordinates $ x _ {1}, \dots, x _ {n} $ (or $ z _ {1}, \dots, z _ {n} $). |
| − | or on the complex space $ \mathbf C ^ {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]]. |
| − | 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==== | ====References==== | ||
Revision as of 01:47, 17 June 2022
$ 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 |
Hessian of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hessian_of_a_function&oldid=47222