Geodesic
Non-negative matrices and their structured singular values
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2023), pp. 36-45.

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In this article, we present new results for the computation of structured singular values of non-negative matrices subject to pure complex perturbations. We prove the equivalence of structured singular values and spectral radius of perturbed matrix $(M\triangle)$. The presented new results on the equivalence of structured singular values, non-negative spectral radius and non-negative determinant of $(M\triangle)$ is presented and analyzed. Furthermore, it has been shown that for a unit spectral radius of $(M\triangle)$, both structured singular values and spectral radius are exactly equal. Finally, we present the exact equivalence between structured singular value and the largest singular value of $(M\triangle)$.
Keywords: $\mu$-values, singular values, eigen values, structured matrices. 
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M. Rehman; T. Rasulov; B. Aminov. Non-negative matrices and their structured singular values. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2023), pp. 36-45. http://geodesic.mathdoc.fr/item/IVM_2023_10_a2/

[1] Doyle J., “Analysis of feedback systems with structured uncertainties”, IEE Proceedings on Control Theory and Applications, 129:6 (1982), 242–250 | DOI | MR

[2] Braatz R.P., Young P.M., Doyle J.C., Morari M., “Computational complexity of $\mu$-calculation”, IEEE Trans. Autom. Control, 39:5 (1994), 1000–1002 | DOI | MR | Zbl

[3] Fan M., Tits A., “Characterization and efficient computation of the structured singular value”, IEEE Trans. Autom. Control, 31:8 (1986), 734–743 | DOI | Zbl

[4] Helton J.W., “A numerical method for computing the structured singular value”, Control Systems control letters, 10 (1988), 21–26 | DOI | MR | Zbl

[5] Young P.M., Newlin M.P., Doyle J.C. et al, “Practical computation of the mixed $\mu$ problem”, American Control Conference, IEEE, 1992, 2190–2194

[6] Young P.M., “The rank one mixed $\mu$ problem and Kharitonov-type analysis”, Automatica, 30:12 (1994), 1899–1911 | DOI | MR | Zbl

[7] Rehman M., Alzabut J., Ateeq T., Kongson J., Sudsutad W., “The Dual Characterization of Structured and Skewed Structured Singular Values”, Math., 10:12 (2022), 2050 | DOI

[8] Guglielmi N., Rehman M., Kressner D., “A Novel Iterative Method to Approximate Structured Singular Values”, SIAM J. Matrix Anal. and Appl., 38:2 (2017), 361–386 | DOI | MR | Zbl

[9] Packard A., Doyle J., “The complex structured singular value”, Automatica, 29:1 (1993), 71–109 | DOI | MR | Zbl

[10] Troeng O., “Five-Full-Block Structured Singular Values of Real Matrices Equal Their Upper Bounds”, IEEE Control Systems Letters, 5:2 (2020), 583–586 | DOI | MR

[11] Rehman M., Tayyab M., Anwar M.F., “Computing $\mu$-Values for Real and Mixed $\mu$ Problems”, Math., 7:9 (2019), 821 | DOI

[12] Berman A., Plemmons R.J., Nonnegative Matrices in the Mathematical Sciences, Classics Appl. Math., SIAM, 1994 | MR | Zbl

[13] Bo Z., “Bounds for the spectral radius of nonnegative matrices”, Math. Slovaca, 51:2 (2001), 179–183 | MR | Zbl

[14] Schur I., Remarks on the theory of bounded bilinear forms with infinitely many variables, Walter de Gruyter, New York, 1911

[15] Nikiforov V., Revisiting Schur's bound on the largest singular value, 2007, arXiv: math/0702722