Page 108 - Textos de Matemática Vol. 39

98 THOMAS LAFFEY AND HELENA SˇMIGOC
spectrum for which a realization by a matrix of the type (3.3) is not possible. A proof that such a representation is always possible would be of great interest.
4. Spectra in which all elements have positive real parts
Assume that we have found realizing matrices Ai for some lists σi, i = 1,...,k, of complex numbers. We can use those matrices to build new nonnegative matrices and realize new lists. To do that we need some informa- tion on the realizing matrices. Here we will present a method that uses the diagonal elements of the matrices Ai. The theorem holds for general [19] as well as for symmetric nonnegative matrices [14].
Theorem 4.1. Let A be an n × n irreducible nonnegative (symmetric) matrix with the Perron eigenvalue λ1, the spectrum (λ1,...,λn) and the diagonal el- ements (a1, a2, . . . , an). Let B be an m × m nonnegative (symmetric) matrix with the Perron eigenvalue μ1, the spectrum (μ1,μ2,...,μm) and the diagonal elements (b1, b2, . . . , bm).
(1) If μ1 ≤ an, then there exists an (n+m−1)×(n+m−1) nonnegative (symmetric) matrix C with the spectrum
(λ1,...,λn,μ2,...,μm)
and the diagonal elements (a1, a2, . . . , an−1, b1, . . . , bm).
(2) If an ≤ μ1, then there exists an (n+m−1)×(n+m−1) nonnegative
symmetric matrix C with the spectrum
(λ1 +μ1 −an,λ2,...,λn,μ2,...,μm)
and the diagonal elements (a1, a2, . . . , an−1, b1, . . . , bm).
Notice that the diagonal elements of the matrix A in the theorem are not ordered. That means that we can choose any diagonal element of the matrix A to play the role of an in the result.
Theorem 4.2 (Sˇmigoc [19]). Let A be a nonnegative n × n matrix with spec- trum σ0 and maximal diagonal element c0. Let Bi, i = 1, . . . , k, be nonnegative mi × mi matrices with maximal diagonal element ci, Perron eigenvalue ρi and spectrum (ρi,σi), where σi is a list of complex numbers. If
c0 +c1 +...+ci−1 ≥ρ1 +ρ2 +...+ρi
for i = 1,...,k, then there exists a nonnegative matrix C with spectrum
(σ0,σ1,...,σk) and maximal diagonal element greater than or equal to c0 +c1 +...+ck −ρ1 −ρ2 −...−ρk.
Suppose σ = (ρ,p1 + iq1,p1 − iq1,...,pr + iqr,pr − iqr), where ρ > 0, pi ≥ 0, i = 1,...,r. We want to find a good estimate for ρ that makes the list σ realizable.