# Computing Nearest Correlation Matrix via Low-Rank ODE’s Based Technique

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## Abstract

**:**

## 1. Introduction

#### Overview of the Article

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 3. Matrix Nearness Problem

## 4. Localization of Smallest Negative Eigenvalue ${\mathbf{\lambda}}_{\mathbf{1}}$

#### 4.1. Construction of Perturbation Matrix

#### 4.2. Formulation of Optimization Problem

#### 4.3. Lemma 4.2.1.

**Proof.**

#### 4.4. A Gradient System of ODE’s

#### 4.5. Characterization of Gradient System of ODE’s

- (i)
- $$\frac{d}{dt}\left({\lambda}_{1}\left(t\right)\right)>0,$$
- (ii)
- $$\dot{E}\left(t\right)=0\iff \frac{d}{dt}\left({\lambda}_{1}\left(t\right)\right)=0,$$
- (iii)
- $$\frac{d}{dt}\left({\lambda}_{1}\left(t\right)\right)=0\iff E\left(t\right)\propto Proj\left({\eta}_{1}{\eta}_{1}^{*}\right).$$

## 5. Localization of ${\mathbf{\lambda}}_{\mathbf{1}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$, ${\mathbf{\lambda}}_{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$

#### 5.1. Optimization Problem

#### 5.2. A Gradient System of ODE’s

**Remark**

**1.**

## 6. Numerical Experimentation

**Example**

**1.**

**Example**

**2.**

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Rehman, M.-U.; Alzabut, J.; Abodayeh, K.
Computing Nearest Correlation Matrix via Low-Rank ODE’s Based Technique. *Symmetry* **2020**, *12*, 1824.
https://doi.org/10.3390/sym12111824

**AMA Style**

Rehman M-U, Alzabut J, Abodayeh K.
Computing Nearest Correlation Matrix via Low-Rank ODE’s Based Technique. *Symmetry*. 2020; 12(11):1824.
https://doi.org/10.3390/sym12111824

**Chicago/Turabian Style**

Rehman, Mutti-Ur, Jehad Alzabut, and Kamaleldin Abodayeh.
2020. "Computing Nearest Correlation Matrix via Low-Rank ODE’s Based Technique" *Symmetry* 12, no. 11: 1824.
https://doi.org/10.3390/sym12111824