complex Hessian matrix

Suppose that f:n be twice differentiableMathworldPlanetmathPlanetmath and let

zk:=12(xk-iyk) and z¯k:=12(xk+iyk).

Then the is the matrix


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) HessianMathworldPlanetmath. 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 continuousMathworldPlanetmath it’s also called a pseudoconvex function).


  • 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