一类有零迹的对称随机矩阵特征值反问题的可解性

Resolubility of Trace Zero Symmetric Stochastic Matrices for the Inverse Eigenvalue Problem

谱σ={λ_1,λ_2,λ_3,0,….0}中至多有3个非零特征值λ_1,λ_2,λ_3,且λ_1≥0≥λ_2≥λ_3,λ_1+λ_2+λ_3=0,在某些特殊情况下,构造n×n阶对称随机矩阵使其以σ为谱的特征值反问题虽已解决,但当n是奇数时,以σ为谱的n×n阶对称随机矩阵是不存在的。

In the special case of where the spectrum zz= |λ1·λ2·λ3·0·. …·0| has at most three nonzero eigenvalies λ1·λ2·λ3 with λ1≥0≥λ2≥λ3,and λ1+λ2+λ3= 0.the inverse eigenvalue problem for symmetric stochastic n × n matrices is solved. It is shown that when n is odd it is not possible to realize the spectrum a with an n × n symmetric stochastic matrix.