与反对合矩阵可交换的对合矩阵

Involutory Matrices Which Are Commutative with Anti-Involutory Matrices

讨论与反对合矩阵可交换的对合矩阵,主要结果如下:(1)与n阶反对合矩阵可交换的对合矩阵的一种表示;(2)对于2阶反对合矩阵A,如果A≠iI(I是单位矩阵),那么与A可交换的对合矩阵一共有4个,它们是±I和±iA;(3)对于3阶反对合矩阵A,如果A≠iI,那么与A可交换的全体对合矩阵为±I和±iA以及±P11k-1P-1,±P-11k-1P-1,P1k l1-k2l-kP-1,P-1k l1-k2l-kP-1其中k是任意复数,l是任意非零复数;当tr(A)=-i时,P是A与diag{i,-i,-i}这一对相似矩阵之间的一个相似因子;当tr(A)=i时,P是A与diag{-i,i,i}之间的一个相似因子。

Abstract： In this paper, we discuss involutory matrices which are commutative with anti-involutory matrices. The main results are as follows： （1）give a kind of representation of the involutory matrices which are commutative with an anti- involutory matrix of order n;（2） for an anti-involutory matrix A of order 2, if A ≠±iI where I is the identity matrix, there are altogether d involutory matrices commutative with A, which are±I and ±iA ; （3） for an anti-involutory matrix A of order 3, if A ≠±iI,the whole involutory matrices commutative with A are ±I and ±iA as well as ±P P^-1,±P P^-1,P P^-1,P P^-1where k is an arbitrary complex number, 1 is an arbitrary nonzero complex number, P is a similar factor between the pair of similar matriceand A and diag{i,-i,-i} if tr（A ）=-i, and that between A and diag{-i, i, i} if tr（A ）=i.