TITLE:
The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation
AUTHORS:
Daniel A. Jaffa
KEYWORDS:
Hessian Matrices, Jacobian Matrices, Laplace Equation, Linear Partial Differential Equations, Systems of Partial Differential Equations, Harmonic Functions, Incompressible and Irrotational Fluid Mechanics
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.12 No.1,
January
26,
2024
ABSTRACT: Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.