Matrix multiplication Definition 7.1.3(Matrix multiplication) Let A=(ay)mxs and B=(by)sxn.Then AB=(cy)mxn,where c可=a1b1y+…+asbg=k=1abg,(i=1,…,m;j=1,…,n)） Special Cases 1.AmxnOnxs =Omxs; OsxmAmxn=Osxn 2.AmxnEn =A; EmAmxn =A. Proof:Let C=AE,then we have: cg=∑aig=agdi=ag: k=1 (Tongji University) 5/14
Matrix multiplication . Definition 7.1.3 (Matrix multiplication) . . Let A = (aij)m×s and B = (bij)s×n . Then AB = (cij)m×n, where cij = ai1b1j + · · · + aisbsj = ∑s k=1 aikbkj , (i = 1, · · · , m; j = 1, · · · , n) . Special Cases . . 1. Am×nOn×s = Om×s , Os×mAm×n = Os×n 2. Am×nEn = A; EmAm×n = A. Proof: Let C = AE, then we have: cij = ∑n k=1 aikδkj = aijδjj = aij. (Tongji University) Linear Algebra 5 / 14
Example 7.1.1 a11 0 0 b11 0 0 0 a22 0 0 b22 0 0 0 ann 0 0 bnn a11b11 0 0 0 a22b22 。。 0 0 0 annbnn Proof: cg=∑a={0 bi ifi=j k=1 fi卡j (Tongji University) 6/14
. Example 7.1.1 .. a11 0 . . . 0 0 a22 . . . 0 ... ... ... 0 0 . . . ann b11 0 . . . 0 0 b22 . . . 0 ... ... ... 0 0 . . . bnn = a11 b11 0 . . . 0 0 a22 b22 . . . 0 ... ... ... 0 0 . . . annbnn . Proof: cij = ∑n k=1 aik bkj = { aii bii if i = j 0 if i ̸= j (Tongji University) Linear Algebra 6 / 14
Some properties (AB)C=A(BC) ⊙入(4B)=入4B=4AB OA(B+C)=AB+AC 8+一- Renmrk 7.11 Note: In general, 4B=4C,4≠0中B=C Tongji University】 7/14
Some properties .1 (AB)C = A(BC) .2 λ(AB) = (λA)B = A(λB) 3. A(B + C) = AB + AC (B + C)A = BA + CA . Remark 7.1.1 . . ( 0 0 a 0 ) ( 1 1 1 1 ) ̸= ( 1 1 1 1 ) ( 0 0 a 0 ) Note: In general, AB = AC, A ̸= 0 ; B = C. A ̸= O , AB = O ; B = O. A ̸= O , B ̸= O ; AB ̸= O. (Tongji University) Linear Algebra 7 / 14
Some properties (AB)C=A(BC) OA(AB)=(入A)B=A(AB) 0B+C=B+0 B+C=BA+C Renmrk 71 Note: In general, 4B=4C,4≠0中B=C Tongji University】 7/14
Some properties .1 (AB)C = A(BC) .2 λ(AB) = (λA)B = A(λB) 3. A(B + C) = AB + AC (B + C)A = BA + CA . Remark 7.1.1 . . ( 0 0 a 0 ) ( 1 1 1 1 ) ̸= ( 1 1 1 1 ) ( 0 0 a 0 ) Note: In general, AB = AC, A ̸= 0 ; B = C. A ̸= O , AB = O ; B = O. A ̸= O , B ̸= O ; AB ̸= O. (Tongji University) Linear Algebra 7 / 14