关于谱半径积r（A）r（AD）对r（A^2 D）的序及其应用
On the Ordering of Spectral Radius Product r(A) r(AD) Versus r(A^2 D) and Related Applications
摘要：
对于非负矩阵A和实对角矩阵D，关于谱半径的两个已知的不等式r（A^2 D^2）>= r（AD）^2和r（A）r（A^2 D^2）>= r（AD）^2，留下了一个问题：是什么决定了r（A^2 D^2）相对于r（A）r（A^2 D^2）的阶。这是生态学和进化动力学中反复出现的一大类问题的一个特例。这里，在A上找到了确定任意方向上的顺序的充分条件。对于特征值全为正且D非负的对角对称非负矩阵A，r（A^2D）<= r（A）r（AD）.如果A的所有特征值除了Perron根之外都是负的，则相反成立。这是一个特殊的情况下，更一般的结果，r（A [（1-m）r（B）I + m B] D）是单调的m时，所有的非Perron特征值的A有相同的符号-减少为正号和增加为负号，对称化非负的A和B交换。交换矩阵包括{ \otimes_{i=1}^L M_i^{t_i}}中的克罗内克积A、B，{0，1，2，.}中的t_i，其包括这些结果的一类应用。该机制允许分析d/d m_j r（{ \otimes_{i=1}^L [（1-m_i）r（A_i）I_i + m_i A_i]}D）的符号。特征值符号条件还提供了马尔可夫链的期望逗留时间的调和平均的下界或上界。这些不平等现象出现在病毒准种的渐近增长率、随机环境中扩散的演化模型以及整个基因组中特定位点突变率的演化中。
英文摘要：
For a nonnegative matrix A and real diagonal matrix D, two known inequalities on the spectral radius, r(A^2 D^2) >= r(AD)^2 and r(A) r(A D^2) >= r(AD)^2, leave open the question of what determines the order of r(A^2 D^2) with respect to r(A) r(A D^2). This is a special case of a broad class of problems that arise repeatedly in ecological and evolutionary dynamics. Here, sufficient conditions are found on A that determine orders in either direction. For a diagonally symmetrizable nonnegative matrix A with all positive eigenvalues and nonnegative D, r(A^2 D) <= r(A) r(AD). The reverse holds if all of the eigenvalues of A are negative besides the Perron root. This is a particular case of the more general result that r(A [(1-m) r(B) I + m B] D) is monotonic in m when all non-Perron eigenvalues of A have the same sign --- decreasing for positive signs and increasing for negative signs, for symmetrizable nonnegative A and B that commute. Commuting matrices include the Kronecker products A, B in {\otimes_{i=1}^L M_i^{t_i}}, t_i in {0, 1, 2, ...}, which comprise a class of application for these results. This machinery allows analysis of the sign of d/d m_j r({\otimes_{i=1}^L[(1-m_i) r(A_i) I_i + m_i A_i]}D). The eigenvalue sign conditions also provide lower or upper bounds to the harmonic mean of the expected sojourn times of Markov chains. These inequalities appear in the asymptotic growth rates of viral quasispecies, models for the evolution of dispersal in random environments, and the evolution of site-specific mutation rates over the entire genome.
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