Eigenvalue regions and the minimal dimension of realization for structured classes of stochastic matrices [r-libre/4299]

Castañeda Betancourt, Juan David, Morneau-Guérin, Frédéric ORCID: https://orcid.org/0000-0001-7610-4648 et Singh, Sarishti (sous presse). Eigenvalue regions and the minimal dimension of realization for structured classes of stochastic matrices. Special Matrices. 10.1515/spma-2025-0067

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Catégorie de document :	Articles de revues
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Étape de publication :	Accepté (sous presse)
Résumé :	Given a class X of stochastic matrices and a complex number lambda, we ask for the smallest dimension of a matrix in the class having lambda as an eigenvalue. We study this problem for several structured classes. For circulant stochastic matrices, the eigenvalue regions are regular polygons, from which we obtain a geometric formula for the minimal realization dimension. We also solve the problem for anticirculant matrices, convex combinations of the identity and a permutation matrix, and convex combinations of powers of a permutation. The latter class connects the realization problem with the Perfect--Mirsky conjecture. For tridiagonal Toeplitz-type stochastic matrices, we show that the eigenvalue region in dimension n is [-cos(pi/n),1] and derive an explicit realization formula. Finally, for pentadiagonal Toeplitz-type matrices, we obtain exact results in dimensions three and four and exhibit a nonreal eigenvalue whose minimal realization dimension is five.
Déposant:	Morneau-Guérin, Frédéric
Responsable :	Frédéric Morneau-Guérin
Dépôt :	23 oct. 2026 15:18
Dernière modification :	26 oct. 2026 11:37

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