# On Spectral Properties of Doubly Stochastic Matrices

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

## 3. Computing Singular Values of Doubly Stochastic Matrices

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 4. Geometrical Interpretation of Spectrum

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 5. Computing Structured Singular Values

#### 5.1. Inner-Algorithm

**Theorem**

**6.**

**Proof.**

#### 5.2. Outer-Algorithm

**Theorem**

**7.**

## 6. Numerical Experimentation

**mussv**.

**Example**

**4.**

**mussv**function.

**mussv**function, the admissible perturbation ∇ is obtained as

**mussv**function approximates the same lower and upper bounds of SSV, i.e., 1.

**Example**

**5.**

**mussv**function.

**mussv**function, the admissible perturbation ∇ is obtained as

**mussv**function approximates the lower bounds as $0.8090$ and upper bound 1 for SSV.

**Example**

**6.**

**mussv**function, the admissible perturbation ∇ is obtained as

**mussv**function approximates the lower bounds as $0.8257$ and upper bound 1 for SSV.

## 7. Conclusions

- The doubly stochastic matrix has an eigenvalue 1.
- The absolute value of any eigenvalue corresponding to a doubly stochastic matrix is less than or equal to 1. The results achieved in this study for structured singular values of doubly stochastic matrices could lead the way to discuss the stability and instability analysis of:
- Stochastic optimal control systems.
- Linear feedback systems in control.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Geometrical interpretation of spectrum and singular values/vectors of Example 1. (

**a**) and (

**b**): spectrum of ${M}_{1},\phantom{\rule{0.166667em}{0ex}}{M}_{2}$ of Example 1, respectively; (

**c**) and (

**d**): geometrical interpretation of the singular values and singular vectors obtained for ${M}_{1},\phantom{\rule{0.166667em}{0ex}}{M}_{2}$ of Example 1, respectively.

**Figure 2.**Geometrical interpretation of spectrum and singular values/vectors of Example 2. (

**a**) and (

**b**): spectrum of ${M}_{3},\phantom{\rule{0.166667em}{0ex}}{M}_{3}$ of Example 2, respectively; (

**c**) and (

**d**): geometrical interpretation of the singular values and singular vectors obtained for ${M}_{3},\phantom{\rule{0.166667em}{0ex}}{M}_{4}$ of Example 2, respectively

**Figure 3.**Geometrical interpretation of spectrum and singular values/vectors of Example 3. (

**a**) and (

**b**): spectrum of Example 3, respectively; (

**c**) and (

**d**): geometrical interpretation of the singular values and singular vectors of Example 3, respectively

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**MDPI and ACS Style**

Rehman, M.-U.; Alzabut, J.; Brohi, J.H.; Hyder, A. On Spectral Properties of Doubly Stochastic Matrices. *Symmetry* **2020**, *12*, 369.
https://doi.org/10.3390/sym12030369

**AMA Style**

Rehman M-U, Alzabut J, Brohi JH, Hyder A. On Spectral Properties of Doubly Stochastic Matrices. *Symmetry*. 2020; 12(3):369.
https://doi.org/10.3390/sym12030369

**Chicago/Turabian Style**

Rehman, Mutti-Ur, Jehad Alzabut, Javed Hussain Brohi, and Arfan Hyder. 2020. "On Spectral Properties of Doubly Stochastic Matrices" *Symmetry* 12, no. 3: 369.
https://doi.org/10.3390/sym12030369