Geodesic
About the spectra of a real nonnegative matrix and its signings
Algebra and discrete mathematics, Tome 32 (2021) no. 1, pp. 1-8.

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For a complex matrix $M$, we denote by $\operatorname{Sp}(M)$ the spectrum of $M$ and by $|M|$ its absolute value, that is the matrix obtained from $M$ by replacing each entry of $M$ by its absolute value. Let $A$ be a nonnegative real matrix, we call a signing of $A$ every real matrix $B$ such that $|B|=A$. In this paper, we characterize the set of all signings of $A$ such that $\operatorname{Sp}(B)=\alpha \operatorname{Sp}(A)$ where $\alpha$ is a complex unit number. Our motivation comes from some recent results about the relationship between the spectrum of a graph and the skew spectra of its orientations.
Keywords: digraphs, irreducible matrices.
Mots-clés : spectra, nonnegative matrices 
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K. Attas; A. Boussaïri; M. Zaidi. About the spectra of a real nonnegative matrix and its signings. Algebra and discrete mathematics, Tome 32 (2021) no. 1, pp. 1-8. http://geodesic.mathdoc.fr/item/ADM_2021_32_1_a0/

[1] A. Anuradha, R. Balakrishnan, X. Chen, X. Li, H. Lian and W. So, “Skew spectra of oriented bipartite graphs”, Electron. J. Combin., 20 (2013), 18 | DOI | MR

[2] A. Anuradha, R. Balakrishnan and W. So, “Skew spectra of graphs without even cycles”, Linear Algebra and its Applications, 444 (2014), 67–80 | DOI | MR | Zbl

[3] N. Biggs, Algebraic Graph Theory, 2nd edition, University Press, Cambridge, 1993 | MR

[4] M. Cavers, S. M. Cioaba, S. Fallat, D. A. Gregory, W. H. Haemers, S. J. Kirkland, J. J. McDonald, M. Tsatsomeros, “Skew-adjacency matrices of graphs”, Linear Algebra Appl., 436 (2012), 4512–4529 | DOI | MR | Zbl

[5] D. Cui and Y. Hou, “On the skew spectra of Cartesian products of graphs”, The Electronic J. Combin., 20:2 (2013), 19 | MR

[6] Y. Hou and T. Lei, “Characteristic polynomials of skew-adjacency matrices of oriented graphs”, The Electronic Journal of Combinatorics, 18 (2011), 156 | DOI | MR

[7] C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, London, 1993 | MR | Zbl

[8] B. Shader and W. So, “Skew spectra of oriented graphs”, The Electronic Journal of Combinatorics, 16 (2009), 32 | DOI | MR

[9] H. Wielandt, “Unzerlegbare, nicht negative Matrizen”, Mathematische Zeitschrift, 52:1 (1950), 642–648 | DOI | MR | Zbl

[10] G. Xu, “Some inequalities on the skew-spectral radii of oriented graphs”, J. Inequal. Appl., 211 (2012), 1–13 | MR | Zbl