与对合矩阵可交换的反对合矩阵

Anti-Involutory Matrices which Are Involutory MatricesCommutative with

讨论与对合矩阵可交换的反对合矩阵。主要结果如下:(1)给出了与n阶对合矩阵可交换的反对合矩阵的一种表示;(2)对于2阶对合矩阵A,如果A≠±I(I是单位矩阵),那么与A可交换的反对合矩阵一共有4个,它们是±iI和±iA;(3)对于3阶对合矩阵A,如果A≠±I,那么与A可交换的全体反对合矩阵为±iI和±iA,以及■其中k是任意复数,l是任意非零复数;当tr(A)=-1时,P是A与diag{1,-1,-1}这一对相似矩阵之间的相似因子;当tr(A)=1时,P是A与diag{-1,1,1}之间的相似因子。

In this paper, we discuss anti-involutory matrices which arc commutative with involutory matriccs.The main results are as follow：（1） give a kind of representation of the anti-involutory matrices which are commutative with an involutory matrix of order n;（2） for an involutory matrix A of order 2, ifA ≠±I where I is the identity matrix, there are altogether 4 anti-involutory matrices commutative with A, which are ±iI and ±iA; （3） for an involutory matrix A of order 3, irA ≠±I,the whole anti-involutory matrices commutative with A are±i Iand ±iA as well as [iik-i]p^-1,±P[-iik-i]P^-1,±P[ikl1＋k^2/l-k]P^-1,P[-ikl-1＋k^2/l-k]P^-1 where k is an arbitrary complex number, l is an arbitrary nonzero complex number, P is a similar factor between the pair of similar malrices A and diag{ 1, - 1, - 1 } if tr（A） = - 1, and that between A and diag{- 1,1,1 } if tr（A ） = 1.