相关矩阵的积和式与特征值

Notes on Nonnegative Correlation Matrices

设A=(a_(ij))是一个nxn非负相关矩阵,=(a_(ij)~2和=(a_(ij)perA(i,j))。本文得出下面两个结论:①(?)的最大特征值λ满足λ≤per(A);②的最大特征值λ满足λ=per(A)。这里per(A)记矩阵A的积和式。

Let A=(a_(ij))be an nxn entrywise nonnegative correlation matrix and (a_(ij)~2),=(a_(ij)perA(i,j)).It is shown that(1)The largest eigenvalue λ of  satisfies λ<per(A).(2)The largest eigenvalue λ of satisfies λ=per(A).Where per(A)denotes the permanent of A.