Matrix operationsBasic notationWe will use the following notations for anm × n matrix A (m rows, n columns).• In terms of the columns of A:A = [a1a2an] =| | |a1a2an| | |• In terms of the entries of A:A =a1,1a1,2a1,na2,1a2,2a2,n am,1am,2am,n, ai,j=entry ini-th row,j-th columnMatrices, just like vectors, are added and scale d componentwise.Example 1.(a)1 05 2+2 33 1=(b) 7 ·2 33 1=Matrix times vectorRecall that (x1, x2,, xn) solves th e l inear system with augmented matrix[A b] =| | | |a1a2anb| | | |if and only ifx1a1+ x2a2++ xnan= b.It is therefore natural to define the product of matrix times vect or asAx = x1a1+ x2a2++ xnan, x =x1xn.Armin [email protected] product of a matrix A with a vector x is a linear combination of the columns ofA with weights given by the entries of x.Example 2.(a)1 05 2·21=(b)2 33 1·01=(c)2 33 1·x1x2=This illustrates that linear systems can be simply expressed as Ax = b:2x1+3x2= b13x1+x2= b22 33 1·x1x2=b1b2Example 3. S u ppose A is m × n and x is in Rp. Under which condition does Ax makesense?Matrix times matrixThe product of matrix times matrix is given byAB = [Ab1Ab2Abp], B = [b1b2bp].Example 4.(a)1 05 2·2 −31 2= because1 05 2·21= and1 05 2·−32= .(b)1 05 2·2 −3 11 2 0= Each column of AB is a linear combination of the colum ns of A with weights givenby the corresponding column of B.Remark 5. The definiti on of the ma tr ix product is inevitable from the mu ltiplication ofmatrix times vector and the fact that we wantAB to be defined such that (AB)x =Armin Strau [email protected](Bx).A(Bx) = A(x1b1+ x2b2+)= x1Ab1+ x2Ab2+= (AB)x if the columns of AB are Ab1, Ab2,Example 6. Suppose A is m × n a nd B is p × q.(a) Under whic h condition does AB ma ke sense?(b) What are the dime nsions of AB in that case?Basic propertiesExample 7.(a)2 33 1·1 00 1=(b)1 00 1·2 33 1=This is the 2 × 2 identity matrix.Theorem 8. Let A, B, C be matrices of appropriate size. Then:•A(BC) = (AB)C associative•A(B + C) = AB + AC left-distributive•(A + B)C = AC + BC right-distributiveExample 9. However, matrix multiplicati on is not commutative!(a)2 33 1·1 10 1=(b)1 10 1·2 33 1=Example 10. Also, a product can be zero even th ough none of the fa c tors is:2 03 0·0 02 1=Armin Strau [email protected] more ways t o look at matrix multiplicationExample 11. What is the entry (AB)i,jat row i and column j?Thej-th column of AB i s A · (col j of B).Rowi of that is (row i of A) · (col j of B). In other words:(AB)i,j= (row i of A) · (col j of B)Use this row-column rule to com pute:2 3 6−1 0 1·2 −30 12 0=Observe the symmetry between r ows and columns in this rule!It follows that the i n t e rpretation“Each column ofA B is a linear combination of the colum ns of A withweights given by the corresp onding column of B.”has the counterp art“Each row of AB is a linear combination of the rows of B with weights givenby the corresponding row of A.”Transpose of a matrixDefinition 12. The transpose ATof a matrix A is the matrix whose columns areformed from the corresponding rows ofA. rows ↔ columnsExample 13.(a)2 03 1−1 4T=(b) [x1x2x3]T=(c)2 33 1T=A matrix A i s called symmetric if A = AT.Armin Strau [email protected] 14. Consider the matri c e sA =1 20 1−2 4, B =1 23 0.Compute:(a) AB =1 20 1−2 41 23 0=(b) (AB)T= (c) BTAT=1 32 01 0 −22 1 4=(d) ATBTW hat’s that fishy sm ell?Theorem 15. Let A, B be matrices of ap propriate size. Then:• (AT)T= A• (A + B)T= AT+ BT• (AB)T= BTATExample 16. Deduce th at (ABC)T= CTBTAT.Questions to check our understanding• True or false?◦ AB has as many column s as B.◦ AB has as many rows as B.Armin Strau
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