Tridiagonal matrices with nonnegative entries

In this paper, we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let d denote a nonnegative integer. Let A denote a matrix in R and let denote the roots of the characteristic polynomial of A. We say A is multiplicity-free whenever these roots are mutually distinct and contained in R. In this case E_i will denote the primitive idempotent of A associated with theta_i(0?i?d). We say A is symmetrizable whenever there exists an invertible diagonal matrix Δ∈R such that ΔAΔ^-1 is symmetric. Let Γ(A) denote the directed graph with vertex set {0,1,…,d}, where i→j whenever i≠j and A_ij≠0.Theorem.Assume that each entry ofAis nonnegative. Then the following are equivalent for0≤s,t≤d.

(i)

The graphΓ(A)is a bidirected path with endpointss,t:s↔*↔*↔?↔*↔t.

(ii)

The matrixAis symmetrizable and multiplicity-free. Moreover the(s,t)-entry ofE_itimes(θ_i-θ₀)?(θ_i-θ_i-1)(θ_i-θ_i+1)?(θ_i-θ_d)is independent of i for0≤i≤d, and this common value is nonzero.

Recently Kurihara and Nozaki obtained a theorem that characterizes the Q-polynomial property for symmetric association schemes. We view the above result as a linear algebraic generalization of their theorem.