內(nèi)容矩陣?yán)碚?5t

1、返回5 M - P廣義逆矩陣A+1定義定義,AGAA GAGGm nn mACGC設(shè)，如果有，使得設(shè)，如果有，使得.MPGAGA 廣義逆廣義逆則稱 是 的矩陣，記為則稱 是 的矩陣，記為(),(),HHGAGA AGAG1定理定理11()()HHHHGDDDBBB ,m nrACABDA 設(shè)是 的最大秩分解 則設(shè)是 的最大秩分解 則.AMPA 就是 的廣義逆矩陣就是 的廣義逆矩陣返回(1)0r 證:證: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的的公公式式法法構(gòu)構(gòu)造造的的右右逆逆B的的公公式式法法構(gòu)構(gòu)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 的廣義逆矩陣是 的廣義逆矩陣11()() ()HHHHHHHAGB BBDDDDB

3、 返回3定理定理m nAC 設(shè)，則有設(shè)，則有(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 設(shè)，則是唯一的設(shè)，則是唯一的返回H TTH TH TTH TBBBDDD 11() () ()()T HTT HT HTT HBBBDDD 11() ()()()()TA (3) ABD 為最大秩分解為最大秩分解()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證:證:(2) ABD 為最大秩分解為最大秩分解,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 的自反廣義逆是 的自反廣義逆()( )rank Arank A ()Hrank A ()()uuuuuuuuuuuuuuu u rHR AR A ()()HR AR A 1111;,HHHAAB DBDDBBDuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu u r u返回(5) AA 是 的自反廣義逆是 的自反廣義逆AAA A 和和是是冪冪等等矩矩陣陣()(),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 的自反廣義=，是 的自反廣義分性：分性：逆矩陣逆矩陣充充( )()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 設(shè)，則有設(shè)，則有(2) ()()() () ;HHHHHAAAAAAA

7、AAA(3)()()() ();HHHHAAAAAAAAAA()()() ().HHHHA AAA AAAAAA證:證: (1) ABD 是最大秩分解是最大秩分解11()()HHHHADDDBBB ()HAA 11()BDHHHHHDBBDDBBDAA 為最大秩分解為最大秩分解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