论文标题
通过perron roots的行根矩阵界限行总和算术平均值
Bounding the row sum arithmetic mean by Perron roots of row-permuted matrices
论文作者
论文摘要
$ r _+^{n \ times n} $表示$ n \ times n $非负矩阵的集合。对于$ a \在r _+^{n \ times n} $中，让$ω（a）$是所有矩阵的集合，可以通过在$ a $ a $的每一行中定位元素来形成。形式上：$$ω（a）= \ {b \ in r _+^{n \ times n}：\ forall i \; \ extists \ extists \ text {a pressuarn} ϕ_i \; \ text {s.t。} \ b_ {i，j} = a_ {i，ϕ_i（j）} \; \ forall j \}。我们表明，$ a $的行总和的算术平均值受$ω（a）$正式的最大和最小光谱半径的限制，我们表明$$ \ min_ {b \inΩ（a）（a）} p（a）} p（a）} p（b）\ leq \ leq \ frac \ frac {1}} a_ {i，j} \ leq \ max_ {b \inΩ（a）}ρ（b）。$$对于正$ a $，我们还为这些不平等之一（或等于同等的，两个）获得了必要的条件，使其成为平等。我们还提供了标准，即使$ρ（c）= \ min_ {b \inΩ（a）}ρ（b）$或$ρ（c）= \ max_ {b \inΩ（a）（a）}ρ（b）$。当所有$ a $的条目均为正时，这些标准用于得出找到此类$ C $的算法。
$R_+^{n\times n}$ denotes the set of $n\times n$ non-negative matrices. For $A\in R_+^{n\times n}$ let $Ω(A)$ be the set of all matrices that can be formed by permuting the elements within each row of $A$. Formally: $$Ω(A)=\{B\in R_+^{n\times n}: \forall i\;\exists\text{ a permutation }ϕ_i\; \text{s.t.}\ b_{i,j}=a_{i,ϕ_i(j)}\;\forall j\}.$$ For $B\inΩ(A)$ let $ρ(B)$ denote the spectral radius or largest non negative eigenvalue of $B$. We show that the arithmetic mean of the row sums of $A$ is bounded by the maximum and minimum spectral radius of the matrices in $Ω(A)$ Formally, we are showing that $$\min_{B\inΩ(A)}ρ(B)\leq \frac{1}{n}\sum_{i=1}^n\sum_{j=1}^n a_{i,j}\leq \max_{B\inΩ(A)}ρ(B).$$ For positive $A$ we also obtain necessary and sufficient conditions for one of these inequalities (or, equivalently, both of them) to become an equality. We also give criteria which an irreducible matrix $C$ should satisfy to have $ρ(C)=\min_{B\inΩ(A)} ρ(B)$ or $ρ(C)=\max_{B\inΩ(A)} ρ(B)$. These criteria are used to derive algorithms for finding such $C$ when all the entries of $A$ are positive .
