内容矩阵理论65t

1、返回5 M - P广义逆矩阵A+1定义定义,AGAA GAGGm nn mACGC设，如果有，使得设，如果有，使得.MPGAGA 广义逆广义逆则称 是 的矩阵，记为则称 是 的矩阵，记为(),(),HHGAGA AGAG1定理定理11()()HHHHGDDDBBB ,m nrACABDA 设是 的最大秩分解 则设是 的最大秩分解 则.AMPA 就是 的广义逆矩阵就是 的广义逆矩阵返回(1)0r 证:证:0A 0A (2)0r ,(,)m rr nrrABD BCDC 存在最大秩分解存在最大秩分解,Hr rHr rrrBBCDDCAGAA GAGG 11()()() ()HHHHHHH HHAG

2、GADDDBBBBD11()()HHHHGDDDBBB D的的公公式式法法构构造造的的右右逆逆B的的公公式式法法构构造造的的左左逆逆11() ()HHHHHHB BBDDDDB 返回11()()HHH HHHB BBDDDDB 11()()HHHHB BBDDDDB 1()HHB BBB 11()()HHHHBDDDDBBB AG 11()() ()HHHHHHHHHGAA GDBB BBDDD11()()HHHHDBB BBDDD 1()HHDDDD 11()()HHHHDDDBBBBD GA GAMPA 是 的广义逆矩阵是 的广义逆矩阵11()() ()HHHHHHHAGB BBDDDDB

3、 返回3定理定理m nAC 设，则有设，则有(1) ();AA (2) ()() ,()();TTHHAAAA(3)()() ;HHHHAAAAAAA(4)()();HR AR A ()()(5),;HR AR AAAPA AP(6)( )().HR AR AAAA A2定理定理.m nACA 设，则是唯一的设，则是唯一的返回H TTH TH TTH TBBBDDD 11() () ()()T HTT HT HTT HBBBDDD 11() ()()()()TA (3) ABD 为最大秩分解为最大秩分解()HHAABDBD HHDBBD ()HHDBBD 11,uuuuuuuuuuuuuuuu

4、uuuuu u rHHBDDBBD11HAAB D ( )()Hrank Arank AAr1()()rrank Drank B()Hrank BBD 1()rank D 11HHAAB DAA 是的是的最大秩分解最大秩分解证:证:(2) ABD 为最大秩分解为最大秩分解,HHBB DD可逆可逆11()()()THHHH TADDDBBB 返回11111111()()()()HHHHHHHAAADD DBBBBD 11() ()()()()HHHHHHHBBDBBD BBDDDD BD 11()()HHHHHHHHDBB BBDDBBDDDDB 1111()()()()HHHHHHHHDBB

5、BBDDBBDDDDB 11()()HHHHDDDBBB A (4) AA 是 的自反广义逆是 的自反广义逆()( )rank Arank A ()Hrank A ()()uuuuuuuuuuuuuuu u rHR AR A ()()HR AR A 1111;,HHHAAB DBDDBBDuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu u r u返回(5) AA 是 的自反广义逆是 的自反广义逆AAA A 和和是是幂幂等等矩矩阵阵()(),R AAR A AAAPA AP = =()( ),()(),()()uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu

6、uuuuuuuuuuuuuuuuuu u ruHR AAR A R A AR AR AR A()(),HR AR AAAPA AP = =(6)AAA AAA=，是 的自反广义=，是 的自反广义分性：分性：逆矩阵逆矩阵充充( )()R AR AA ()R A A = =()R A ()HR A ()(),( )()HR AR AHR AR AAAPA AP 必要性：必要性：uuuuuuuuuuuuuuuuuuuuuuuuuuu u rAAA A 返回(1) ()() ,()();HHHHAAAAAAAA4定理定理m nAC 设，则有设，则有(2) ()()() () ;HHHHHAAAAAAA

7、AAA(3)()()() ();HHHHAAAAAAAAAA()()() ().HHHHA AAA AAAAAA证:证: (1) ABD 是最大秩分解是最大秩分解11()()HHHHADDDBBB ()HAA 11()BDHHHHHDBBDDBBDAA 为最大秩分解为最大秩分解uuuuuuuuuuuuuuuuuuuuuuuu1442uuu r44 3uu u u11()()()HHHHHHBBDBBDDBBDDD ()HAA 返回1111()()()()HHHHHHDBB BBDDBBDDD 1111()()()()HHHHHHDDDBBBB BBDDD 1111()() ()()HHHHHHDDDBBBB BBDDD ()HAA (2) ()()HHAAAA ()HAA () HHHAAAA () HHAAAA ()HAAAA (3)() HHAAAAAA ()()( )HHAAAAI 返回() ()() ()HHHHHHAAAABDDBBDDB 11()()HHHHHHHHBDDDDBBDDBBBBDDB 1111()()()()HHHHHHHHBDDDDB BDDB BB BDD B 111()()()HHHHHHBDDDDBBDDDDB 11()()HHHHBD DDDBBB ()AAII ()()HHAAAA () ()HHAAAAAA