{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:46:04Z","timestamp":1760237164228,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T00:00:00Z","timestamp":1583107200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012639","name":"Prince Sultan University","doi-asserted-by":"publisher","award":["RG-DES-2017-01-17"],"award-info":[{"award-number":["RG-DES-2017-01-17"]}],"id":[{"id":"10.13039\/501100012639","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The relationship among eigenvalues, singular values, and quadratic forms associated with linear transforms of doubly stochastic matrices has remained an important topic since 1949. The main objective of this article is to present some useful theorems, concerning the spectral properties of doubly stochastic matrices. The computation of the bounds of structured singular values for a family of doubly stochastic matrices is presented by using low-rank ordinary differential equations-based techniques. The numerical computations illustrating the behavior of the method and the spectrum of doubly stochastic matrices is then numerically analyzed.<\/jats:p>","DOI":"10.3390\/sym12030369","type":"journal-article","created":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T07:50:53Z","timestamp":1583135453000},"page":"369","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Spectral Properties of Doubly Stochastic Matrices"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5828-3133","authenticated-orcid":false,"given":"Mutti-Ur","family":"Rehman","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sukkur IBA University, Sukkur 65200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5262-1138","authenticated-orcid":false,"given":"Jehad","family":"Alzabut","sequence":"additional","affiliation":[{"name":"Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6315-9105","authenticated-orcid":false,"given":"Javed Hussain","family":"Brohi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sukkur IBA University, Sukkur 65200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Arfan","family":"Hyder","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sukkur IBA University, Sukkur 65200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gnacik, M., and Kania, T. (2019). Inverse problems for symmetric doubly stochastic matrices whose suleimanova spectra spectra are to be bounded below by 1\/2. arXiv.","DOI":"10.1016\/j.laa.2020.01.029"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1017\/S0305004100027626","article-title":"On positive stochastic matrices with real characteristic roots","volume":"Volume 48","author":"Perfect","year":"1952","journal-title":"Mathematical Proceedings of the Cambridge Philosophical Society"},{"key":"ref_3","first-page":"35","article-title":"Methods of constructing certain stochastic matrices","volume":"69","author":"Perfect","year":"1965","journal-title":"Duke Math. J."},{"key":"ref_4","first-page":"343","article-title":"Stochastic matrices with real characteristic values","volume":"66","author":"Suleimanova","year":"1949","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"ref_5","first-page":"33","article-title":"The question of a necessary and sufficient condition for the existence of a stochastic matrix with prescribed characteristic numbers","volume":"28","author":"Suleimanova","year":"1965","journal-title":"Trudy Vsesojuz. Zaocn. Energet. Inst. Vyp"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"462","DOI":"10.13001\/1081-3810.1325","article-title":"On spectra perturbation and elementary divisors of positive matrices","volume":"18","author":"Ccapa","year":"2009","journal-title":"Electron. J. Linear Algebra"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/S0024-3795(03)00366-5","article-title":"The inverse eigenvalue problem for symmetric doubly stochastic matrices","volume":"379","author":"Hwang","year":"2004","journal-title":"Linear Algebra Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/j.laa.2015.11.033","article-title":"Perron spectratopes and the real nonnegative inverse eigenvalue problem","volume":"493","author":"Johnson","year":"2016","journal-title":"Linear Algebra Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.laa.2013.12.005","article-title":"On the symmetric doubly stochastic inverse eigenvalue problem","volume":"445","author":"Lei","year":"2014","journal-title":"Linear Algebra Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1382","DOI":"10.1016\/j.laa.2013.04.023","article-title":"An algorithm for constructing doubly stochastic matrices for the inverse eigenvalue problem","volume":"439","author":"Mourad","year":"2013","journal-title":"Linear Algebra Appl."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Berman, A., and Plemmons, R.J. (1994). Non-Negative Matrices in the Mathematical Sciences, SIAM Publications.","DOI":"10.1137\/1.9781611971262"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Bhatia, R. (1997). Matrix Analysis, Springer.","DOI":"10.1007\/978-1-4612-0653-8"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Seneta, E. (1981). Non-Negative Matrices and Markov Chains, Springer. [2nd ed.].","DOI":"10.1007\/0-387-32792-4"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3400","DOI":"10.1016\/j.laa.2011.11.034","article-title":"On a spectral property of doubly stochastic matrices and its application to their inverse eigenvalue problem","volume":"436","author":"Mourad","year":"2012","journal-title":"Linear Algebra Appl."},{"key":"ref_15","unstructured":"Mourad, B. (1998). Generalized Doubly-Stochastic Matrices and Inverse Eigenvalue Problems, University of New South Wales."},{"key":"ref_16","unstructured":"Mehlum, M. (2012). Doubly Stochastic Matrices and the Assignment Problem. [Master\u2019s Thesis, University of Oslo]."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1080\/03081080600899296","article-title":"On a conjecture about the eigenvalues of doubly stochastic matrices","volume":"55","author":"Mashreghi","year":"2007","journal-title":"Linear Multilinear Algebra"},{"key":"ref_18","first-page":"853","article-title":"Some results on the symmetric doubly stochastic inverse eigenvalue problem","volume":"43","author":"Xu","year":"2017","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Young, P.M., and Doyle, J.C. (1990, January 7). Computation of mu with real and complex uncertainties. Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, HI, USA.","DOI":"10.1109\/CDC.1990.203804"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Young, P.M., Newlin, M.P., and Doyle, J.C. (1992, January 26). Practical computation of the mixed \u03bc problem. Proceedings of the 1992 American Control Conference, Chicago, IL, USA.","DOI":"10.23919\/ACC.1992.4792521"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/BFb0043286","article-title":"Robustness in the presence of parametric uncertainty and unmodeled dynamics","volume":"130","author":"Fan","year":"1989","journal-title":"Adv. Comput. Control."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1137\/16M1074977","article-title":"A novel iterative method to approximate structured singular values","volume":"38","author":"Guglielmi","year":"2019","journal-title":"SIAM J. Matrix Anal. 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