同济大学：《线性代数》课程教学资源（课件讲稿）Matrices

Matrices Li Zhang Department of Mathematics Tongji University 4907 Tongji University】 1/14

. . Matrices Li Zhang Department of Mathematics Tongji University (Tongji University) Linear Algebra 1 / 14

Matrix Definition 7.0.1 a11 aml amn Note:m x n matrix,square matrix,entry,order,main diagonal,row matrix, column matrix The negative of a matrix is the matrix whose entries are the negatives of the original matrix. A zero matrix is one of any size all of whose entries are 0. Tongji University】 2/14

Matrix . Definition 7.0.1 . .   a11 . . . a1n . . . . . . am1 . . . amn   Note: m × n matrix, square matrix, entry, order, main diagonal, row matrix, column matrix The negative of a matrix is the matrix whose entries are the negatives of the original matrix. A zero matrix is one of any size all of whose entries are 0. (Tongji University) Linear Algebra 2 / 14

Matrix Definition 7.0.1 a11 aml amn Note:m x n matrix,square matrix,entry,order,main diagonal,row matrix, column matrix The negative of a matrix is the matrix whose entries are the negatives of the original matrix. A zero matrix is one of any size all of whose entries are 0. Tongji University】 2/14

Matrix . Definition 7.0.1 . .   a11 . . . a1n . . . . . . am1 . . . amn   Note: m × n matrix, square matrix, entry, order, main diagonal, row matrix, column matrix The negative of a matrix is the matrix whose entries are the negatives of the original matrix. A zero matrix is one of any size all of whose entries are 0. (Tongji University) Linear Algebra 2 / 14

Matrix addition Definition 7.1.1 (Matrix addition) a11...a1n b11..b1n a11+b11.a1n+b1n 十 .. = am1...amn bm1...bmn aml bmi...amn bmn 04十B=B十 ©H+B)+C=过+(B+C 0A+B一C=4-C+B 04十B=C+D=4+B-C=D Tongji University】 3/14

Matrix addition . Definition 7.1.1 (Matrix addition) . .   a11 . . . a1n . . . . . . am1 . . . amn   +   b11 . . . b1n . . . . . . bm1 . . . bmn   :=   a11 + b11 . . . a1n + b1n . . . . . . am1 + bm1 . . . amn + bmn   .1 A + B = B + A. .2 (A + B) + C = A + (B + C) 3. A + B − C = A − C + B .4 A + B = C + D ⇒ A + B − C = D (Tongji University) Linear Algebra 3 / 14

Matrix addition Definition 7.1.1 (Matrix addition) a11...a1n b11..b1n a11+b11.a1n+b1n 十 三 am1...amn bm1...bmn aml十bml..amm+bmm OA+B=B+A. ⊙(A+B)+C=A+(B+C) A+B-C=A-C+B OA+B=C+D→A+B-C=D Tongji University】 3/14

Matrix addition . Definition 7.1.1 (Matrix addition) . .   a11 . . . a1n . . . . . . am1 . . . amn   +   b11 . . . b1n . . . . . . bm1 . . . bmn   :=   a11 + b11 . . . a1n + b1n . . . . . . am1 + bm1 . . . amn + bmn   .1 A + B = B + A. .2 (A + B) + C = A + (B + C) 3. A + B − C = A − C + B .4 A + B = C + D ⇒ A + B − C = D (Tongji University) Linear Algebra 3 / 14

Matrix addition Definition 7.1.1 (Matrix addition) a11...a1n b11..b1n a11+b11.a1n+b1n 十 三 am1...amn bm1...bmn aml十bml..amm+bmm OA+B=B+A. ⊙(A+B)+C=A+(B+C) A+B-C=A-C+B OA+B=C+D→A+B-C=D Tongji University】 3/14

Matrix addition . Definition 7.1.1 (Matrix addition) . .   a11 . . . a1n . . . . . . am1 . . . amn   +   b11 . . . b1n . . . . . . bm1 . . . bmn   :=   a11 + b11 . . . a1n + b1n . . . . . . am1 + bm1 . . . amn + bmn   .1 A + B = B + A. .2 (A + B) + C = A + (B + C) 3. A + B − C = A − C + B .4 A + B = C + D ⇒ A + B − C = D (Tongji University) Linear Algebra 3 / 14

scalar multiplication The word"scalar"just means"number"-we use a separate word to distinguish numbers from other mathematical objects we want to calculate with,like matrices. Definition 7.1.2 (scalar multiplication) To multiply a matrix by a scalar,multiply every entry of the matrix by that scalar. a11 入a11 Xain ·.. aml amn 入aml 入amn 0A04)=A4 0十)4=人4+ OA4+B=入H+AB Tongji University】 4114

scalar multiplication . The word ”scalar” just means ”number” – we use a separate word to distinguish numbers from other mathematical objects we want to calculate with, like matrices. . Definition 7.1.2 (scalar multiplication) . . To multiply a matrix by a scalar, multiply every entry of the matrix by that scalar. λ   a11 . . . a1n . . . . . . am1 . . . amn   :=   λa11 . . . λa1n . . . . . . λam1 . . . λamn   .1 λ(µA) = (λµ)A .2 (λ + µ)A = λA + µA 3. λ(A + B) = λA + λB (Tongji University) Linear Algebra 4 / 14

scalar multiplication The word"scalar"just means"number"-we use a separate word to distinguish numbers from other mathematical objects we want to calculate with,like matrices. Definition 7.1.2 (scalar multiplication) To multiply a matrix by a scalar,multiply every entry of the matrix by that scalar. a11 入a11 Xain aml .。amn 入am1l 入amn Oλ(4)=(4)4 ⊙(A+4)A=入M+4 OX(A+B)=入A+入B Tongji University】 4/14

scalar multiplication . The word ”scalar” just means ”number” – we use a separate word to distinguish numbers from other mathematical objects we want to calculate with, like matrices. . Definition 7.1.2 (scalar multiplication) . . To multiply a matrix by a scalar, multiply every entry of the matrix by that scalar. λ   a11 . . . a1n . . . . . . am1 . . . amn   :=   λa11 . . . λa1n . . . . . . λam1 . . . λamn   .1 λ(µA) = (λµ)A .2 (λ + µ)A = λA + µA 3. λ(A + B) = λA + λB (Tongji University) Linear Algebra 4 / 14

scalar multiplication The word"scalar"just means"number"-we use a separate word to distinguish numbers from other mathematical objects we want to calculate with,like matrices. Definition 7.1.2 (scalar multiplication) To multiply a matrix by a scalar,multiply every entry of the matrix by that scalar. a11 入a11 Xain aml .。amn 入am1l 入amn Oλ(4)=(4)4 ⊙(A+4)A=入M+4 OX(A+B)=入A+入B Tongji University】 4/14

scalar multiplication . The word ”scalar” just means ”number” – we use a separate word to distinguish numbers from other mathematical objects we want to calculate with, like matrices. . Definition 7.1.2 (scalar multiplication) . . To multiply a matrix by a scalar, multiply every entry of the matrix by that scalar. λ   a11 . . . a1n . . . . . . am1 . . . amn   :=   λa11 . . . λa1n . . . . . . λam1 . . . λamn   .1 λ(µA) = (λµ)A .2 (λ + µ)A = λA + µA 3. λ(A + B) = λA + λB (Tongji University) Linear Algebra 4 / 14

Matrix multiplication Definition 7.1.3(Matrix multiplication) Let A=(ay)mxs and B=(bi)sxn.Then AB=(cy)mxn.where cg=a1b1y+…十asbg=k=1 aikbij,(=1,…,m;j=1,…,n） Special Cases IAmxnOnxs=Omx Osxm4mx=0xxn 24m×:= m行= Proof:Let C=AE.then we have 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