上海海事大学矩阵论必考题-陈建彪

1.设A∈C m×n, B∈C p×q, D∈C, X∈C n×p, 则矩阵方程

AXB = D

有解的充分必要条件是对某一个A( 1 ), B( 1 )使AA( 1 )D B( 1 )B = D 成立, 且矩阵方程通解为

X = A( 1 )D B( 1 )+ ( Z - A( 1 )AZBB( 1 )) ,

Z 为任意n×p 矩阵.

证: 必要性AXB = D 有解, 两边左乘AA( 1 ) ,右乘B( 1 )B, 可得:

AA( 1 )AXBB( 1 ) B = AA( 1 )DB( 1 ) B,

即证得: D = AA( 1 )D B( 1 )B .

充分性因A( A( 1 )D B( 1 ) ) B = D, 可见X0 = A( 1 )D B( 1 )是AX B= D 的解. 下证通解为X = A( 1 ) D B( 1 )+ ( Z - A( 1 )AZBB( 1 ) ) .

AXB = A( A( 1 ) D B( 1 )) B + A( Z - A( 1 )AZBB( 1 )) B =

AA( 1 ) D B( 1 ) B + AZB - AA( 1 )AZBB( 1 )B =

D + AZB - AZB = D .

故X = A( 1 ) D B( 1 ) + ( Z - A( 1 ) AZBB( 1 ) ) 是该方程之解.

设Y 为AXB = D 的任一解:即AYB = D, 则

Y = Y + A( 1 ) D B( 1 )- A( 1 ) D B( 1 )=

Y + A( 1 )D B( 1 )- A( 1 ) AYBB( 1 )( 令Y = Z) =

A( 1 ) D B( 1 )+ ( Z - A( 1 ) AZBB( 1 ) ) .