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

## Abstract

**:**

## 1. Introduction

## 2. More Solutions to Quadratic Equations

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 3. Semi-Antinorm Related Complex Numbers

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**1.**

**Remark**

**1.**

- (a)
- Because of the assumption (6), those φ-angles that appear in this theorem do not attain values that are multiples of $\pi /2$.
- (b)
- In conclusion, let us name the invariance property that the semi-antinorm ${|.|}_{p}$-value of a vector in ${\mathbb{R}}^{2}$ is not changed when multiplied by a vector which has semi-antinorm ${|.|}_{p}$-value one. In other words, for every $r>0,$ the Lie group on ${C}_{p}\left(r\right)$ consists of all transformations ${(x,y)}^{T}\to {(x,y)}^{T}{\odot}_{p}z$ where z is an arbitrary element of ${C}_{p}\left(1\right).$
- (c)
- The particular vector p-multiplication of a vector $z\in {\mathbb{R}}^{2}$ by $\mathfrak{e}={(1,0)}^{T}$ results in vector z.

**Definition**

**4.**

## 4. Elliptical Complex Numbers

#### 4.1. Vector Analysis

**Definition**

**5.**

**Definition**

**6.**

**Example**

**5.**

**Definition**

**7.**

**Theorem**

**2.**

**Proof.**

#### 4.2. Complex Analysis

**Definition**

**8.**

**Example**

**6.**

#### 4.3. Geometric View

**Theorem**

**3.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

## 5. Complex Numbers Related to Matrix Homogeneous Functionals

**Definition**

**9.**

**Theorem**

**4.**

**Proof.**

**Remark**

**4.**

**Definition**

**10.**

## 6. Invariant Probability Densities

**Example**

**7.**

**Example**

**8.**

**Example**

**9.**

## 7. Concluding Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Geometry of quadratic equations with respect to standard vector multiplication. The Euclidean circles have the radii $\sqrt{n}=1.5$ and $n=2.25$, respectively.

**Figure 2.**Geometry of quadratic equations with respect to ${l}_{p}$-vector multiplication. The exemplary drawn p-circles (with $1<p<2$) have radii of the sizes $\sqrt{n}=1.5$ and $n=2.25$, similar figures can be drawn for $p\in (0,1]$ and $p>2$.

**Figure 3.**Level one lines of norm (blue), antinorm (pink) and semi-antinorm (black) ${\left|z\right|}_{p}$.

**Figure 4.**Level r lines of the semi-antinorm $z\to {\left|z\right|}_{-1}$ and level one lines of the semi-antinorms ${\left|z\right|}_{p}$ .

**Figure 6.**The p-generalized sine and cosine functions, $p<0$. For figures in case $p>0$, see [7]. The jumps that are seen in multiples of $\pi /2$ occur from +1 to −1, or vice versa.

**Figure 7.**Vector p-products $w=u{\odot}_{-1}v$ and $z=z1\phantom{\rule{0.277778em}{0ex}}{\odot}_{-2/3}\phantom{\rule{0.277778em}{0ex}}z2$.

**Figure 9.**$[p,q]$-circles ${C}_{[p,q]}\left(r\right)$ for smaller and larger $[p,q]$-radii show change of shape and orientation as r changes.

**Figure 11.**Vector $[p,q]$-products $w=u{\odot}_{[3,\phantom{\rule{0.277778em}{0ex}}0.7]}v$ and $z3=z1{\odot}_{[1.7,\phantom{\rule{0.277778em}{0ex}}0.3]}z2$.

**Figure 12.**Power exponential density $\varphi (x,y)={C}_{[p,q]}\xb7{e}^{-{\left|x\right|}^{p}/p-{\left|y\right|}^{q}/q}$ for particular values of $[p,q]$.

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**MDPI and ACS Style**

Richter, W.-D. Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals. *Axioms* **2021**, *10*, 340.
https://doi.org/10.3390/axioms10040340

**AMA Style**

Richter W-D. Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals. *Axioms*. 2021; 10(4):340.
https://doi.org/10.3390/axioms10040340

**Chicago/Turabian Style**

Richter, Wolf-Dieter. 2021. "Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals" *Axioms* 10, no. 4: 340.
https://doi.org/10.3390/axioms10040340