Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals

Abstract

We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities.

Keywords

• vector multiplication
• vector division
• vector exponential function
• generalized trigonometric functions
• complex algebraic structure
• generalized complex plane