讲分块矩阵的概念及运算

1、1ijr kr AB( , ( )i j kEA=BirAB ( ( )EA = Bi ijrrAB ( , )EA = Bi jijccAB ( , )AEBi jicAB( ( )AEBi jickc AB( , ( )i j kAEB 对一般情况我们有对一般情况我们有: :(0) (0) 2A B一次行一次行存在相应初等阵存在相应初等阵P 使使:PA=BA B一次列一次列存在相应初等阵存在相应初等阵Q使使:AQ=B 初等阵初等阵与与初等变换初等变换的关系的关系3A B行行存在存在初等阵初等阵P1 P2Ps使使: :P1P2PsA=BA B列列存在存在初等阵初等阵Q1 Q2Qt使使: :A

2、Q1Q2Qt=B存在存在初等阵初等阵P1 P2Ps Q1 Q2Qt使使: :P1P2PsAQ1Q2Qt=B A B（ A B ）初初初等阵初等阵与与初等变换初等变换的关系的关系41kAPP1,kPP A E初初1111111111AP PP EQ QQsstt11,ijPQ 初等阵初等阵.即存在初等即存在初等 ,P PP Q QQ1212stP1P2PsAQ1Q2Qt=E( )5PAQBn,PQA B 初初A B 1212,stPPPP QQQQ,P PP Q QQ1212stP1P2Ps AQ1Q2Qt=BPAQB6()()()( )r PAr AQr PAQr AA E行行. . A1A使

3、使,12P PPs112AP PPs1A A12P PP AEs7m n.000rEPAQ1.kAPP111.kPP A E111.kAPPEPAQB()()()( )r PAr AQr PAQr A8由由A可逆可逆，有有1sPPA Enn可得可得11sPPEA9n (2n)1A EE A1AEEA10021112111A11 0 21 1 0 01 12 0 1 01 1 1 0 0 1 AE11 3511 112022A ,1 0 01352220 1 0 1112220 0 1 011可验证可验证1AAE1 1 2 0 1 00 21 1 0 00 0 1 0 1 11211121321

4、2223313233,aaaAaaaaaa212223111213311132123313aaaBaaaaa aa aa,12010100100010001101PP13 ,AXBA：1()()A BE A B行行AXB,515853323912100AB可验证可验证A可逆可逆.14()A B515853323912100XAB114253610014010250013615 16aaaaA aaaaaaaa111213142122232431323334 A 视为一个子块视为一个子块 ija视为一个子块视为一个子块AAAA1112212217aaaaA aaaaaaaa11121314212

5、2232431323334aaaaAaaaaaaaa1112131421222324313233341234 12318sAAAA1212*sAAAA19 20()()()ij s tij s tijij s tA BABAB()()ijs tijs tkkkAAA21AB =C (Am p , Bp n ),11111111trsstttrAABBABAABBijijijittjCA BA BA B1122, ;, .tikkjkA Bis jr11122EEAB222102010002001000200321000030722212EA B020231402123tsstAAAAA1111

6、TTT()ijAATTTTststAAAA1111TTTttAAAA11tAA124Ai，12msAAA,12mmmsAAA12sAAA()mN.sAAA122512*,sAAAAAi，12sAA AAA0 (1,2, )iAis2611121sAAA1112sAAAA1112sAAAA11111ssAAA当当 0 (1,2, )iAisAAi，27,A1000420000340001求求AA1及及A 100042000034000110 3442 012 36设设28A1110004200003400011000120021400330001111 004 23 400 129设设,3,2.

7、3 A,BMAB3(1)50AABB求求35AB333( 5) AB432 353(2)50AABB3 3( 1)3AAB333( 1) AA B52 3 3 33( 1)50ABAB30(,)12nABA B BB( , , )0 00* *A为一子为一子块块m pp nAB0(,)12nAB ABAB, .01jABjn31(,(,)121nABA B BB( ,)12nCC CC( )mmAABAABABBAA B11222mCCCC12,1iiABC im(,)12nAB ABABm pp nm nABC,1jjABCjn32 33ABMCD34 ()ijijrrcc(),iiPrc

8、PP(),ijjirKrcc KK35( )00200mnPEEP或或( )030mmnnEEKKEE或或( )00100mnmnmnEEEEEE或或 36 EABECD00 ABECDE00CDABBADC37 PABECD00 EABKECD0ABPCDE00ABECDKE0PA PBCDAPBCPDABK A C K B DABKBCDKD38 分块分块()A E1()E A分块分块AE1EA39 ,A，B .ACB10mnA CEBE000mnEA CAEB111000行行mnEAA CBEB1111000行行行行mnEA CABE11000A CAA CBBB111110040A B

9、( )AABB11100010( )ABBA11100200( )A CAA CBBB11111003( )AACBB CAB1111100441其中其中A、B为为n阶方阵阶方阵.A B| |,ABA B0 nAEB行行0 nABEB0nAEB0 nABEB()201nnnABBEAB()201nnABBE0nABBE0nABE42A CD BACACDBBDAC10|A BDA C1ACBDA C10证明证明 其中其中A 为为n阶阶可逆矩阵可逆矩阵, ,B为为m阶阶方阵方阵. .|ACA BDA CDB1( (行列式第一降阶定理行列式第一降阶定理) )43 证明证明|Em -AB| = |En -BA| ,其中其中 A 为为mn阶阶矩阵矩阵, , B为为nm阶阶阵阵. .mnEABE|mnnEAEBABE0 mnEAEBA行行mnEABE0 mnEABBE行行|mmnEAEABBE44 mEAB1 mmEAB1 mnEBA m nnEBA(0) 0 | |mnEABEBA, , m nmnEABEBA m n为任意数为任意数.45mnn46nnnnnnnn nnnn nxyxyxyxyxyxyxyxyxyxyxyxyEx yx yx yx yx yx y1 11 211 11 212 12 222 12