关于置换矩阵的最小多项式

On the Least Polynomial of Permutation Matrix

设π(S_i)是一个S_i×S_i循环置换阵,[λ^(s1)-1,…,λ^(st-1)-1,λ^(st)-1]表示λ^(s1)-1,…,λ^(st-1)-1,λ^(st)-1表示的最小公倍式。本文首先指出,任何一个n×n置换矩阵P是相似于矩阵 diag(I_k,π(S_1),…,π(S_1),…,π(S_t),…,π(S_t))的,这里k+sum from i=1 to t (k_iS_i)=n。之后我们证明了P的最小多项式 m_p(λ)=[λ^(s1)-1,…,λ^(st-1)-1,λ^(st)-1]。

Let (S_t) be a S_t×S_t cyclic permutation matrix, [λ^(S_1)-1,…, λ^(S_t-1)-1, λ^(S_t)-1] denotes the least common multiple of λ^(S_1)-1,…,λ^(S_t-1)-1, and λ^(S_t)-1. This paper first points out that any n×n permutation matrix P is similar to matrix Diag (I,π(S_1),…,π(S_1),…,π(S_t),…,π (S_t)) where k+sum from i=1 to t(k_i S_i=n). Then it proves that the least polynomial of p is m_P(λ)=[λ^(S_1)-1,…,λ^(S_t-1)-1,λ^(S_t)-1].