## Spectral properties of one class of sign-symmetric matrices

#### Citation

Kushel, O. Y. (2009). Spectral properties of one class of sign-symmetric matrices. Maltepe Üniversitesi. s. 312.#### Abstract

A matrix A of a linear operator A : R
n → R
n is called J –sign-symmetric, if there exists such a subset J ⊆ {1, . . . , n},
that the inequality aij ≤ 0 follows from the inclusions i ∈ J , j ∈ J c
and j ∈ J , i ∈ J c
for any two numbers i, j, and one
of the inclusions i ∈ J , j ∈ J c
or j ∈ J , i ∈ J c
follows from the strict inequality aij < 0 (here J
c = {1, . . . , n} \ J ).
This definition is a generalization of the well-known definition of positive matrices, which are widely used in economics,
mechanics, biology and other branches of science.
Let A be a J –sign-symmetric matrix, and let J be a subset of {1, . . . , n} in the definition of J –sign-symmetricity.
Let its second compound matrix A(2) also be a J –sign-symmetric matrix. Let Je be a subset of {1, . . . , C2
n} in the
definition of J –sign-symmetricity for the matrix A(2). Let us construct the set W(J , Je) ⊆ ({1, . . . , n} × {1, . . . , n})
by the following way: (i, j) ∈ W(J , Je) if and only if one of the following two cases takes place:
(a) both the numbers i, j belong either to the set J , or to the set J
c
, besides, if i < j, then the number of the pair
(i, j) in the lexicographic numeration belongs to the set Je, and if i > j, then the number of the pair (j, i) belongs to the
set Jec = {1, . . . , C2
n} \ Je;
(b) one of the numbers i, j belongs to the set J , the other belongs to the set J
c
, besides, if i < j, then the number of
the pair (i, j) belongs to the set Jec
, and if i > j, then the number of the pair (j, i) belongs to the set Je.
Such a set is not uniquely defined, but there is a finite number of different ways of its constructing. The set W(J , Je)
is called transitive if the inclusion (i, k) ∈ W(J , Je) follows from the inclusions (i, j) ∈ W(J , Je) and (j, k) ∈ W(J , Je) for
any indices i, j, k ∈ {1, . . . , n}.
Theorem 0.1 Let the matrix A of a non-zero linear operator A be J –sign-symmetric together with its second compound
matrix A(2). Then the operator A has a positive eigenvalue λ1 = ρ(A). More than that, if λ1 is a simple eigenvalue,
then one of the following two cases takes place:
(1) If at least one of the possible sets W(J , Je) is transitive, then the second in modulus eigenvalue λ2 of the operator
A is nonnegative and different in modulus from the first eigenvalue λ1.
(2) If all the possible sets W(J , Je) are not transitive, there there is an odd number k of eigenvalues on the spectral
circle |λ| = ρ(A). All of them are simple and coincide with kth roots of (ρ(A))k
.

#### Source

International Conference of Mathematical Sciences#### Collections

- Makale Koleksiyonu [586]

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