Very recently, Qi and Cui extended the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts and proved that they have Perron eigenpair and Perron-Frobenius eigenpair. The Collatz method was also extended to find Perron eigenpair. Qi and Cui proposed two conjectures. One is the k-order power of a dual number matrix tends to zero if and only if the spectral radius of its standard part less than one, and another is the linear convergence of the Collatz method. In this paper, we confirm these conjectures and provide theoretical proof. The main contribution is to show that the Collatz method R-linearly converges with an explicit rate.

翻译：近日，Qi与Cui将Perron-Frobenius理论推广至标准部分为本原不可约非负矩阵的对偶数矩阵，证明其存在Perron特征对及Perron-Frobenius特征对。同时，Collatz方法也被推广用于求解Perron特征对。Qi与Cui提出两个猜想：其一，对偶数矩阵的k次幂趋于零当且仅当其标准部分的谱半径小于1；其二，Collatz方法具有线性收敛性。本文证实了这两个猜想并给出理论证明，主要贡献在于证明Collatz方法以显式速率实现R线性收敛。