Editing Hessian matrix (section)

=== Generalization to the complex case ===

In the context of [[several complex variables]], the Hessian may be generalized.  Suppose <math>f\colon\Complex^n \to \Complex,</math> and write <math>f\left(z_1, \ldots, z_n\right).</math>  Identifying <math>{\mathbb{C}}^n</math> with <math>{\mathbb{R}}^{2n}</math>, the normal "real" Hessian is a <math>2n \times 2n</math> matrix. As the object of study in several complex variables are [[Holomorphic function|holomorphic functions]], that is, solutions to the n-dimensional [[Cauchy–Riemann equations|Cauchy–Riemann conditions]], we usually look on the part of the Hessian that contains information invariant under holomorphic changes of coordinates.  This "part" is the so-called complex Hessian, which is the matrix <math>\left(\frac{\partial^2f}{\partial z_j \partial\bar{z}_k}\right)_{j,k}.</math>  Note that if <math>f</math> is holomorphic, then its complex Hessian matrix is identically zero, so the complex Hessian is used to study smooth but not holomorphic functions, see for example [[Pseudoconvexity|Levi pseudoconvexity]].  When dealing with holomorphic functions, we could consider the Hessian matrix <math>\left(\frac{\partial^2f}{\partial z_j \partial z_k}\right)_{j,k}.</math>