双非负矩阵幂的临界指数猜想
The critical exponent conjecture for powers of doubly nonnegative matrices
摘要：
双非负矩阵在包括马尔可夫随机场（正带状图模型）和马尔可夫链的收敛性分析中自然出现。在这篇短文中，我们解决了C.R. Johnson等人[线性代数Appl.435（2011）]通过证明$n$ × $n$双非负矩阵的所有连续常规幂在其之上都是双非负的临界指数恰好是$n-2$。我们表明，猜想如下立即应用一般的表征从文献。我们证明了一个更强的形式的猜想，通过分类所有的权力保持双非负矩阵，并继续推广的猜想广泛的功能类。我们还提供了不同的方法来解决原来的猜想。
英文摘要：
Doubly non-negative matrices arise naturally in many setting including Markov random fields (positively banded graphical models) and in the convergence analysis of Markov chains. In this short note, we settle a recent conjecture by C.R. Johnson et al. [Linear Algebra Appl. 435 (2011)] by proving that the critical exponent beyond which all continuous conventional powers of $n$-by-$n$ doubly nonnegative matrices are doubly nonnegative is exactly $n-2$. We show that the conjecture follows immediately by applying a general characterization from the literature. We prove a stronger form of the conjecture by classifying all powers preserving doubly nonnegative matrices, and proceed to generalize the conjecture for broad classes of functions. We also provide different approaches for settling the original conjecture.
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