# 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

## References

- Gnacik, M.; Kania, T. Inverse problems for symmetric doubly stochastic matrices whose suleimanova spectra spectra are to be bounded below by 1/2. arXiv
**2019**, arXiv:1909.01291. [Google Scholar] [CrossRef] - Perfect, H. On positive stochastic matrices with real characteristic roots. In Mathematical Proceedings of the Cambridge Philosophical Society; Cambridge University Press: Cambridge, UK, 1952; Volume 48, pp. 271–276. [Google Scholar]
- Perfect, H. Methods of constructing certain stochastic matrices. Duke Math. J.
**1965**, 69, 35–57. [Google Scholar] [CrossRef] - Suleimanova, H.R. Stochastic matrices with real characteristic values. Dokl. Akad. Nauk SSSR
**1949**, 66, 343–345. [Google Scholar] - Suleimanova, H.R. The question of a necessary and sufficient condition for the existence of a stochastic matrix with prescribed characteristic numbers. Trudy Vsesojuz. Zaocn. Energet. Inst. Vyp
**1965**, 28, 33–49. [Google Scholar] - Ccapa, J.; Soto, R.L. On spectra perturbation and elementary divisors of positive matrices. Electron. J. Linear Algebra
**2009**, 18, 462–481. [Google Scholar] [CrossRef][Green Version] - Hwang, S.-G.; Pyo, S.-S. The inverse eigenvalue problem for symmetric doubly stochastic matrices. Linear Algebra Appl.
**2004**, 379, 77–83. [Google Scholar] [CrossRef][Green Version] - Johnson, C.R.; Paparella, P. Perron spectratopes and the real nonnegative inverse eigenvalue problem. Linear Algebra Appl.
**2016**, 493, 281–300. [Google Scholar] [CrossRef] - Lei, Y.-J.; Xu, W.-R.; Lu, Y.; Niu, Y.-R.; Gu, X.-M. On the symmetric doubly stochastic inverse eigenvalue problem. Linear Algebra Appl.
**2014**, 445, 181–205. [Google Scholar] [CrossRef] - Mourad, B.; Abbas, H.; Mourad, A.; Ghaddar, A.; Kaddoura, I. An algorithm for constructing doubly stochastic matrices for the inverse eigenvalue problem. Linear Algebra Appl.
**2013**, 439, 1382–1400. [Google Scholar] [CrossRef] - Berman, A.; Plemmons, R.J. Non-Negative Matrices in the Mathematical Sciences; SIAM Publications: Philadelphia, PA, USA, 1994. [Google Scholar]
- Bhatia, R. Matrix Analysis; Springer: New York, NY, USA, 1997. [Google Scholar]
- Seneta, E. Non-Negative Matrices and Markov Chains, 2nd ed.; Springer: New York, NY, USA, 1981. [Google Scholar]
- Mourad, B. On a spectral property of doubly stochastic matrices and its application to their inverse eigenvalue problem. Linear Algebra Appl.
**2012**, 436, 3400–3412. [Google Scholar] [CrossRef][Green Version] - Mourad, B. Generalized Doubly-Stochastic Matrices and Inverse Eigenvalue Problems; University of New South Wales: Sydney, Australia, 1998. [Google Scholar]
- Mehlum, M. Doubly Stochastic Matrices and the Assignment Problem. Master’s Thesis, University of Oslo, Oslo, Norway, 2012. [Google Scholar]
- Mashreghi, J.; Rivard, R. On a conjecture about the eigenvalues of doubly stochastic matrices. Linear Multilinear Algebra
**2007**, 55, 491–498. [Google Scholar] [CrossRef] - Xu, W.-R.; Chen, G.-L. Some results on the symmetric doubly stochastic inverse eigenvalue problem. Bull. Iran. Math. Soc.
**2017**, 43, 853–865. [Google Scholar] - Young, P.M.; Doyle, J.C. Computation of mu with real and complex uncertainties. In Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, HI, USA, 7 December 1990; pp. 1230–1235. [Google Scholar]
- Young, P.M.; Newlin, M.P.; Doyle, J.C. Practical computation of the mixed μ problem. In Proceedings of the 1992 American Control Conference, Chicago, IL, USA, 26 June 1992; pp. 2190–2194. [Google Scholar]
- Fan, M.K.H.; Doyle, J.C.; Tits, A.L. Robustness in the presence of parametric uncertainty and unmodeled dynamics. Adv. Comput. Control.
**1989**, 130, 363–367. [Google Scholar] - Guglielmi, N.R.; Mutti, U.; Kressner, D. A novel iterative method to approximate structured singular values. SIAM J. Matrix Anal. Appl.
**2019**, 38, 361–386. [Google Scholar] [CrossRef][Green Version]

**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