关于矩阵乘积的结构和谱的一些结果
Some results on the structure and spectra of matrix-products
摘要：
我们考虑某些矩阵乘积，其中乘积中的连续矩阵交替地属于特定的定性类或其转置。主要定理将这些矩阵积的结构和谱性质与底层二部图的结构联系起来。一个结果是毛毛虫的一个特征：一个图是毛毛虫当且仅当所有的矩阵产品与它有真正的非负谱。其他几个等价的这种证明。这项工作的灵感来自于某些问题，在动力系统中，这些产品自然产生的雅可比矩阵，结果有影响的存在性和稳定性的平衡在这些系统中。
英文摘要：
We consider certain matrix-products where successive matrices in the product belong alternately to a particular qualitative class or its transpose. The main theorems relate structural and spectral properties of these matrix-products to the structure of underlying bipartite graphs. One consequence is a characterisation of caterpillars: a graph is a caterpillar if and only if all matrix-products associated with it have real nonnegative spectrum. Several other equivalences of this kind are proved. The work is inspired by certain questions in dynamical systems where such products arise naturally as Jacobian matrices, and the results have implications for the existence and stability of equilibria in these systems.
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