XXV COMCA On the spectral radius under certain

XXV COMCA On the spectral radius under certain

XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y de Agosto de 2016, Antofagasta, Chile
On the spectral radius under certain perturbations, 2
Miriam Pisonero∗
Departamento de Matemática Aplicada
Universidad de Valladolid/IMUVA
Valladolid, Spain
Abstract
It is well known that increasing (perturbing) an element of a nonnegative matrix A nondecreases
its spectral radius ρ(A), and that it increases when A is irreducible. Or equivalently, if the weight
of an arc of a weighted digraph D is increased (perturbed), then the spectral radius ρ(D) does
not decrease, and it increases when D is strongly connected.
Let Eij be the elementary matrix: 1 in the (i, j)-position and 0’s in all other positions. In
general, it is difficult to characterize when the spectral radius of a nonnegativeP
matrix A, n × n,
is smaller than or equal to the spectral radius of a perturbed matrix A + i,j δij Eij , with
δij ≥ 0. We study the particular case when a principal submatrix of A is perturbed:
X
A+
δij Eij , {k1 < · · · < kt } ⊂ {1, . . . , n}
i,j∈{k1 <···<kt }
and we give explicit conditions of polynomial type. Similar comments and results are pplied to
the perturbation of the weight function of a weighted digraph.
Joint work with:
C. Marijuán1 , Dpto. Matemática Aplicada, Universidad de Valladolid/IMUVA, Valladolid, Spain.
References
[1] C. R. Johnson, C. Marijuán, M. Pisonero, Submatrix monotonicity of the Perron root,
Linear Algebra and its Applications 437, 2012, pp. 2429-2435.
[2] S. Furtado, C. R. Johnson, C. Marijuán, M. Pisonero, Submatrix monotonicity of the
Perron root, II, Linear Algebra and its Applications 458, 2014, pp. 679-688.
∗ e-mail:
[email protected]
supported by MTM2015-365764-C3-1-P, e-mail: [email protected]
1 Partially
1