# Examples on the Non-Uniqueness of the Rank 1 Tensor Decomposition of Rank 4 Tensors

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

**Theorem**

**2.**

**Conjecture**

**1.**

**Theorem**

**3.**

**Proposition**

**1.**

**Remark**

**1.**

## 2. Preliminaries

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

#### 2.1. Linear Projections

## 3. Existence Results for Type I and Type II Tensors

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Definition**

**3.**

**Proposition**

**4.**

**Proof.**

## 4. $\mathbf{Y}={\mathbb{P}}^{\mathbf{3}}\times {\mathbb{P}}^{\mathbf{3}}\times {\mathbb{P}}^{\mathbf{1}}$

**Proposition**

**5.**

**Proof.**

**Remark**

**5.**

**Theorem**

**4.**

**Proof.**

**Claim**

**1.**

**Proof.**

**Observation**

**1.**

**Claim**

**2.**

**Proof.**

**Claim**

**3.**

**Proof.**

**Proposition**

**6.**

**Proof.**

## 5. Examples

**Proposition**

**7.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 6. End of the Proofs

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**3.**

**Proof**

**of**

**Proposition**

**1.**

**Remark**

**6.**

## 7. Effectiveness and Further Questions

**Remark**

**7.**

**Remark**

**8.**

**Remark**

**9.**

**Definition**

**4.**

**Open**

**Problem**

**2.**

## 8. Methods and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Ballico, E.
Examples on the Non-Uniqueness of the Rank 1 Tensor Decomposition of Rank 4 Tensors. *Symmetry* **2022**, *14*, 1889.
https://doi.org/10.3390/sym14091889

**AMA Style**

Ballico E.
Examples on the Non-Uniqueness of the Rank 1 Tensor Decomposition of Rank 4 Tensors. *Symmetry*. 2022; 14(9):1889.
https://doi.org/10.3390/sym14091889

**Chicago/Turabian Style**

Ballico, Edoardo.
2022. "Examples on the Non-Uniqueness of the Rank 1 Tensor Decomposition of Rank 4 Tensors" *Symmetry* 14, no. 9: 1889.
https://doi.org/10.3390/sym14091889