The spectra of nonnegative integer matrices via formal power series
Ki Kim, Nicholas Ormes, F.W. Roush · Journal of the American Mathematical Society · 2000
We characterize the possible nonzero spectra of primitive integer matrices (the integer case of Boyle and Handelman’s Spectral Conjecture). Characterizations of nonzero spectra of nonnegative matrices over Z {\mathbb Z} and Q {\mathbb Q} follow from this result. For the proof of the main theorem we use polynomial matrices to reduce the problem of realizing a candidate spectrum ( λ 1 , λ 2 , … , λ d ) (\lambda _1,\lambda _2,\ldots ,\lambda _d) to factoring the polynomial ∏ i = 1 d ( 1 − λ i t ) \prod _{i=1}^d (1-\lambda _it) as a product ( 1 − r ( t ) ) ∏ i = 1 n ( 1 − q i ( t ) ) (1-r(t))\prod _{i=1}^n (1-q_i(t)) where the q i q_i ’s are polynomials in t Z + [ t ] t{\mathbb Z}_+[t] satisfying some technical conditions and