Note on the Fundamental Theorem on Irreducible Non-Negative Matrices
Note on the Fundamental Theorem on Irreducible Non-Negative Matrices
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DOI：
10.1017/s0013091500010816
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发表时间：
1958-11
影响因子：
0.7
通讯作者：
H. Schneider
中科院分区：
文献类型：
--
作者：
H. Schneider
设A=[aij]是n阶不可约非负矩阵。众所周知，矩阵A有一个正的特征根ρ(假设n>；L)，它在每个特征根λ满足|λ|≤ρ的意义上是简单和最大的，并且属于ρ的特征向量x可以选择为正。这些结果最初是由Frobenius得出的，但Wielandt(4)用一个极其简单的基本概念证明了这些结果。最近，Household给出了Wielandt证明的一个变体(2)。
Let A = [aij] be an n-th order irreducible non-negative matrix. As is very well-known, the matrix A has a positive characteristic root ρ (provided that n>l), which is simple and maximal in the sense that every characteristic root λ satisfies |λ| ≤ρ, and the characteristic vector x belonging to ρ may be chosen positive. These results, originally due to Frobenius, have been proved by Wielandt (4) by means of a strikingly simple basic idea. Recently, a variant of Wielandt's proof has been given by Householder (2).