# complex Hessian matrix

Suppose that $f:{\u2102}^{n}\to \u2102$ be twice differentiable^{}
and let

$$\frac{\partial}{\partial {z}_{k}}:=\frac{1}{2}\left(\frac{\partial}{\partial {x}_{k}}-i\frac{\partial}{\partial {y}_{k}}\right)\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\frac{\partial}{\partial {\overline{z}}_{k}}:=\frac{1}{2}\left(\frac{\partial}{\partial {x}_{k}}+i\frac{\partial}{\partial {y}_{k}}\right).$$ |

Then the is the matrix

$$\left[\begin{array}{cccc}\hfill \frac{{\partial}^{2}f}{\partial {z}_{1}\partial {\overline{z}}_{1}}\hfill & \hfill \frac{{\partial}^{2}f}{\partial {z}_{1}\partial {\overline{z}}_{2}}\hfill & \hfill \mathrm{\dots}\hfill & \hfill \frac{{\partial}^{2}f}{\partial {z}_{1}\partial {\overline{z}}_{n}}\hfill \\ \hfill \frac{{\partial}^{2}f}{\partial {z}_{2}\partial {\overline{z}}_{1}}\hfill & \hfill \frac{{\partial}^{2}f}{\partial {z}_{2}\partial {\overline{z}}_{2}}\hfill & \hfill \mathrm{\dots}\hfill & \hfill \frac{{\partial}^{2}f}{\partial {z}_{2}\partial {\overline{z}}_{n}}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill \frac{{\partial}^{2}f}{\partial {z}_{n}\partial {\overline{z}}_{1}}\hfill & \hfill \frac{{\partial}^{2}f}{\partial {z}_{n}\partial {\overline{z}}_{2}}\hfill & \hfill \mathrm{\dots}\hfill & \hfill \frac{{\partial}^{2}f}{\partial {z}_{n}\partial {\overline{z}}_{n}}\hfill \end{array}\right].$$ |

When applied to tangent vectors of the zero set of $f$,
it is called the Levi form and used to define a Levi
pseudoconvex point of a boundary of a domain. Note that the
matrix is not the same as the (real) Hessian^{}. A twice continuously
differentiable real valued
function with a
positive semidefinite real Hessian matrix at every point is convex, but a function with
positive semidefinite matrix at every point is
plurisubharmonic (since it’s
continuous^{} it’s also called a pseudoconvex function).

## References

- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | complex Hessian matrix |
---|---|

Canonical name | ComplexHessianMatrix |

Date of creation | 2013-03-22 14:31:16 |

Last modified on | 2013-03-22 14:31:16 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 7 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32-00 |

Related topic | HessianMatrix |