內容矩陣理論65t

1、返回5 M - P廣義逆矩陣A+1定義定義,AGAA GAGGm nn mACGC設，如果有，使得設，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 設是 的最大秩分解 則設是 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構造造的的右右逆逆B的的公公式式法法構構造造的的左左逆逆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，則有設，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，則是唯一的設，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，則有設，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