关于两个幂等矩阵组合群逆的探讨

关于两个幂等矩阵组合群逆的探讨 Discussions on the Group Inverses of Combinations of Two Idempotent Matrices first-author 曹秋红，女，硕士生，现从事线性代数及其应用方面的研究. E-mail:1501645519@qq.com 曹秋红，女，硕士生，现从事线性代数及其应用方面的研究. E-mail: 1501645519@qq.com 1501645519@qq.com corresp 谢涛, E-mail:xietao_1294@163.com 谢涛, E-mail:xietao_1294@163.com xietao_1294@163.com 湖北师范大学 数学与统计学院, 湖北 黄石 435002 湖北师范大学 数学与统计学院, 湖北 黄石 435002 College of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, Hubei, China College of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, Hubei, China 运用矩阵零空间的性质证明了复数域上两个不同的非零幂等矩阵 P , Q 的组合 a 1 P + b 1 Q + a 2 PQ + b 2 QP +…+ a 2 n －1 ( PQ ) n －1 P + b 2 n －1 ( QP ) n －1 Q + a 2 n ( PQ ) n (其中 a 1 , b 1 , …, b 2 n －1 , a 2 n ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle , a 1 , b 1 ≠0)在条件( QP ) n =0( n ≥2)下的秩与系数的选取无关，进而证明了其群逆存在.另外, 还得到了组合 a P + b Q +c PQ + d QP 在条件( QP ) n =0下的群逆表达式. By using the properties of null space of matrices, the rank of the combinations a 1 P + b 1 Q + a 2 PQ + b 2 QP +…+ a 2 n －1 ( PQ ) n －1 P + b 2 n －1 ( QP ) n －1 Q + a 2 n ( PQ ) n of two different nonzero idempotent matrices P and Q over the complex field http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle , where a 1 , b 1 , …, a 2 n ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle , a 1 , b 1 ≠0, was proved to be independent with the choice of its coefficients and under the condition ( QP ) n =0( n ≥2). Therefore, the existence of the group inverse of the combination was also obtained. In addition, the formula for the group inverse of the combination a P + b Q + c PQ + d QP was presented under the condition ( QP ) n =0. 幂等矩阵 线性组合 群逆 idempotent matrix linear combination group inverse 引言 1 若 P ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle n × n 且满足 P 2 = P , 则称 P 是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle 上的幂等矩阵或者投射子.幂等矩阵是一类重要且性质非常丰富的矩阵，在线性算子理论、统计学、几何学、控制理论中有重要应用 [ 1 1 ~ 4 4 1~4 ] .近年来, 中外学者对各种特殊矩阵的组合的相关性质进行了广泛和深入地研究.文献[ 5 5 ]证明了复数域上两个非零的幂等矩阵 P , Q 线性组合 a P + b Q ( a , b , a + b ≠0)的可逆性与系数的选取无关.文献[ 6 6 ]进一步得到了 a P + b Q ( a , b , a + b ≠0)的秩与系数的选取无关.随后, 研究人员发现对于Hilbert空间上的有界幂等线性算子 P , Q , a P + b Q ( a , b , a + b ≠0)的值域, 可逆性, Fredholm性等也有类似的上述性质 [ 7 7 ~ 10 10 7~10 ] .这些研究工作启发了人们对各种特殊矩阵及算子线性组合的性质与其系数关系的深入探讨 [ 11 11 , 12 12 ] . 2009年, 文献[ 13 13 , 14 14 ]分别考虑了幂等矩阵 P , Q 的组合 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258524&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258524&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258524&type=middle 与 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258533&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258533&type=middle 的可逆性与系数的关系，推广了前人的结果.随后, 对上述组合的各种广义逆的存在性、广义逆的表达式与系数的关系引起了热烈的讨论 [ 15 15 ~ 20 20 15~20 ] . 2011年, 文献[ 21 21 ]证明了若 P , Q 是复数域上的两个不同的非零幂等矩阵, 且满足( QP ) 2 =0, 则组合 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258552&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258552&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258552&type=middle 的群逆存在, 其中 a , b ≠0.另外, 文献[ 21 21 ]还得到了组合(1)的群逆表达式. 在上述研究的基础上，一个问题自然地被提出来，即若 P , Q 是复数域上的两个不同的非零幂等矩阵, 且满足( QP ) n =0或( PQ ) n =0( n ≥2), 则组合 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258563&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258563&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258569&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258569&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258569&type=middle 的群逆是否存在, 其中 a 1 , b 1 ≠0？若组合(2)和(3)群逆存在, 其群逆表达式是怎样的？ 本文运用矩阵零空间的性质证明了复数域上两个非零的幂等矩阵 P , Q 的组合(2)和(3)分别在条件( QP ) n =0, ( PQ ) n =0( n ≥2)下的秩与系数的选取无关，并进而证明了其群逆存在.另外，还得到了组合 a P + b Q + c PQ + d QP 在条件( QP ) n =0或( PQ ) n =0下的群逆表达式.本文证明组合(2)的群逆存在性的方法与文献[ 21 21 ]不同. 预备知识 1 设 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle n × n 是复数域上所有 n 阶矩阵构成的集合, 对于 A ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle n × n , 若存在 X ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle n × n 使得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258577&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258577&type=middle 成立，则称 X 为 A 的一个群逆.若 A 存在群逆, 则其群逆唯一, 记 A 的群逆为 A g .对于 A ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle n × n , A g 存在当且仅当 r ( A 2 )= r ( A ), 其中 r ( A )表示 A 的秩. 引理1 [ 21 21 ]  设 P , Q 是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle n × n 上的两个不同的非零幂等矩阵, 且满足( QP ) 2 =0, 则对任意的 a , b , c , d , e , f , g ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle , 其中 a , b ≠0, 组合 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258582&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258582&type=middle 的群逆存在, 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258590&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258590&type=middle 容易举例说明( PQ ) 2 =0并不意味着( QP ) 2 =0, 反之也成立. 引理2 [ 21 21 ]  设 P , Q 是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle n × n 上的两个不同的非零幂等矩阵, 且满足( PQ ) 2 =0, 则对任意的 a , b , c , d , e , f , h ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle , 其中 a , b ≠0, 组合 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258596&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258596&type=middle 的群逆存在, 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258602&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258602&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258602&type=middle 例1  令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258604&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258604&type=middle 则可得到 P , Q 是幂等矩阵, 且满足( QP ) 2 ≠0, ( QP ) 3 =0.另外, 若 P , Q 满足( QP ) 2 =0, 则( QP ) 3 =( QP ) 2 QP =0.因此条件( QP ) 3 =0比条件( QP ) 2 =0弱.条件( PQ ) n =0比条件( PQ ) n －1 =0 ( n ≥2)弱.另外, 条件( PQ ) n =0与( QP ) n =0之间也没有蕴含关系.所以在条件( QP ) n =0或( PQ ) n =0下讨论组合(2)和(3)群逆的存在性及表达式是有意义的. 主要结论 1 下面将证明复数域上两个非零幂等矩阵 P , Q 的组合(2)和(3)分别在条件( QP ) n =0, ( PQ ) n =0( n ≥2)下的秩与系数的关系，进而讨论其群逆的存在性. 定理1  设 P , Q 是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle n × n 上的两个不同的非零幂等矩阵, Φ 为(2)式，其中 a i , b j ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle (1≤ i ≤2 n , 1≤ j ≤2 n －1)且 a 1 b 1 ≠0.若( QP ) n =0, 则以下结论成立: 1) N ( Φ )= N ( P )∩ N ( Q )； 2) r ( Φ )是一个常数且 r ( Φ ) = r ( P + Q ), 从而 Φ 可逆当且仅当 P + Q 可逆； 3) Φ 存在群逆. 其中 N ( A )表示矩阵 A 的核空间, 即 N ( A ) = { x http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle n | Ax =0}. 证   1)一方面, 设∀ α ∈ N ( P )∩ N ( Q ), 则有 P α = Q α =0, 所以 Φ α =0, 因此 N ( P )∩ N ( Q ) ⊆ N ( Φ ).另一方面, 设⊆ α ∈ N ( Φ ), 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258609&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258609&type=middle 在(4)式两边左乘( QP ) n －1 Q , 并注意到( QP ) n =0可得 b 1 ( QP ) n －1 Q α =0. 因为 b 1 ≠0, 所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258614&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258614&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258614&type=middle 在(4)式两边右乘( PQ ) n , 并注意到( QP ) n =0可得 a 1 ( PQ ) n α =0. 因为 a 1 ≠0, 所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258623&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258623&type=middle 在(4)式两边左乘( QP ) n －1 , 并注意(5)式，则 b 1 ( QP ) n －1 α =0, 可得( QP ) n －1 α =0.依次在(4)式两边左乘( QP ) n －2 Q , ( QP ) n －2 , ( QP ) n －3 Q, ( QP ) n －3 , …, QP , Q , 可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258629&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258629&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258629&type=middle 同理依次在(4)式两边左乘 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258638&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258638&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258638&type=middle 可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258643&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258643&type=middle 从而 α ∈ N ( P )∩ N ( Q ).所以 N ( Φ )⊆ N ( P )∩ N ( Q ).因此 N ( Φ ) = N ( P )∩ N ( Q ). 即结论1)成立. 2) 因为 r ( Φ ) = n －dim( N ( Φ )) = n －dim( N ( P )∩ N ( Q ))是一个常数, 所以 r ( Φ )与满足条件 a 1 , b 1 ≠0的系数 a i , b j ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle (1≤ i ≤2 n , 1≤ j ≤2 n －1)选取无关.特别地, 取 a 1 = b 1 =1, a i = b j =0(1≤ i ≤2 n , 1≤ j ≤2 n －1), 则 r ( Φ ) = r ( P + Q ).从而 r ( Φ )可逆当且仅当 P + Q 可逆. 3) 注意到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258649&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258649&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258649&type=middle 且 Φ 2 的系数 a 1 2 ≠0, b 1 2 ≠0, 即 Φ 2 的系数满足定理的条件, 由结论1)和2)知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258652&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258652&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258652&type=middle 从而 Φ 的群逆存在. 定理2  设 P , Q 是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle n × n 上的两个不同的非零幂等矩阵, Φ 为(3)式，其中 a i , b j ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle (1≤ i ≤2 n , 1≤ j ≤2 n －1)且 a 1 b 1 ≠0.若( PQ ) n =0, 则以下结论成立: 1) N ( Φ ) = N ( P )∩ N ( Q )； 2) r ( Φ )是一个常数且 r ( Φ ) = r ( P + Q ), 从而 Φ 可逆当且仅当 P + Q 可逆； 3) Φ 存在群逆. 证  类似定理1的证明. 下面考虑组合 a P + b Q + c PQ + d QP 在条件( QP ) n 或( PQ ) n 下的群逆公式. 定理3  设 a , b , c , d ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle , 且满足 a ≠0, b ≠0, A = a P + b Q + c PQ + d QP , 其中 P , Q 是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle n × n 上两个不同的非零幂等矩阵，且满足( QP ) n =0, 则 A 存在群逆, 且表达式为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258655&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258655&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258655&type=middle 其中, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258658&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258658&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258658&type=middle 证  令 X = X 1 + X 2 + X 3 , 其中, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258661&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258661&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258661&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258668&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258668&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258668&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258676&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258676&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258676&type=middle 由( QP ) n =0可知, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258684&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258684&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258684&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258692&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258692&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258692&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258698&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258698&type=middle 所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258713&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258713&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258713&type=middle 又因为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258718&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258718&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258718&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258721&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258721&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258727&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258727&type=middle 所以, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258750&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258750&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258750&type=middle 因此 AX = XA . 又因为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258758&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258758&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258758&type=middle 所以 AXA = A , XAX = X . 故由群逆的惟一性知， X 为 A 的群逆. 定理4  设 a , b , c , d ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle , 且满足 a ≠0, b ≠0, A = a P + b Q + c PQ + d QP , 其中 P , Q 是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle n × n 上两个不同的非零幂等矩阵，且满足( PQ ) n =0, 则 A 存在群逆, 且表达式为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258762&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258762&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258762&type=middle 其中, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258765&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258765&type=middle 证  类似定理3的证明. 结论 1 本文讨论了复数域上两个不同的非零幂等矩阵 P , Q 的组合 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258770&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258770&type=middle (其中 a 1 , b 1 , …, b 2 n －1 , a 2 n ∈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258880&type=middle , a 1 , b 1 ≠0)在条件( QP ) n =0( n ≥2)下的秩与系数的关系及其群逆的存在性，还研究了组合 a P + b Q + c PQ + d QP 在条件( QP ) n =0下的群逆的计算公式.研究上述组合在其他条件下的广义逆的存在性及表达式是一个可以继续讨论的问题. 参考文献 CAMPBELL S L, MEYER C D. 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Linear and Multilinear Algebra , 2011, 59 (1):101-115. 10.14188/j.1671-8836.2018.03.010 O151.2 whdxxblxb-64-3-262 湖北省教育厅青年项目(B2017149);湖北师范大学博士科研启动项目(B201603);湖北师范大学研究生科研创新项目(20170116) 2017-06-20 2018-06-24 2020-11-30 版权所有© 2018 《武汉大学学报(理学版)》编辑部 Copyright © 2018 Journal of Wuhan University(Natural Science Edition). All rights reserved. 2018 武汉大学学报(理学版) 3 64