An upper bound on the characteristic polynomial of a nonnegative matrix leading to a proof of the Boyle–Handelman conjecture (original) (raw)

In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if A A is an ( n + 1 ) × ( n + 1 ) (n+1)\times (n+1) nonnegative matrix whose nonzero eigenvalues are: λ 0 ≥ | λ i | \lambda _0 \geq |\lambda _i| , i = 1 , … , r i=1,\ldots ,r , r ≤ n r \leq \ n , then for all x ≥ λ 0 x \geq \lambda _0 , ( ∗ ) ∏ i = 0 r ( x − λ i ) ≤ x r + 1 − λ 0 r + 1 . \begin{equation} \prod _{i=0}^{r} (x-\lambda _i) \leq x^{r+1}-\lambda _0^{r+1}.\tag *{$(\ast )$} \end{equation} To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when 2 ( r + 1 ) ≥ ( n + 1 ) 2(r+1)\geq (n+1) , while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when n ≤ 4 n\leq 4 and when the spectrum of A A is real. They also showed that the conjecture is asymptotically true with the dimension. Here we prove a slightly stronger inequality than in ( ∗ ) (\ast ) , from which it follows tha...