{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 18 0 0 0 0 0 0 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"Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 264 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 265 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 266 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 267 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 268 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 269 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 270 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 271 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 272 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 273 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 268 20 "Maple Linear Algebra" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 68 "Start any session where you plan on using matr ices with the command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "w ith(linalg);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^r%.BlockDiagonalG%,G ramSchmidtG%,JordanBlockG%)LUdecompG%)QRdecompG%*WronskianG%'addcolG%' addrowG%$adjG%(adjointG%&angleG%(augmentG%(backsubG%%bandG%&basisG%'be zoutG%,blockmatrixG%(charmatG%)charpolyG%)choleskyG%$colG%'coldimG%)co lspaceG%(colspanG%*companionG%'concatG%%condG%)copyintoG%*crossprodG%% curlG%)definiteG%(delcolsG%(delrowsG%$detG%%diagG%(divergeG%(dotprodG% *eigenvalsG%,eigenvaluesG%-eigenvectorsG%+eigenvectsG%,entermatrixG%&e qualG%,exponentialG%'extendG%,ffgausselimG%*fibonacciG%+forwardsubG%*f robeniusG%*gausselimG%*gaussjordG%(geneqnsG%*genmatrixG%%gradG%)hadama rdG%(hermiteG%(hessianG%(hilbertG%+htransposeG%)ihermiteG%*indexfuncG% *innerprodG%)intbasisG%(inverseG%'ismithG%*issimilarG%'iszeroG%)jacobi anG%'jordanG%'kernelG%*laplacianG%*leastsqrsG%)linsolveG%'mataddG%'mat rixG%&minorG%(minpolyG%'mulcolG%'mulrowG%)multiplyG%%normG%*normalizeG %*nullspaceG%'orthogG%*permanentG%&pivotG%*potentialG%+randmatrixG%+ra ndvectorG%%rankG%(ratformG%$rowG%'rowdimG%)rowspaceG%(rowspanG%%rrefG% *scalarmulG%-singularvalsG%&smithG%,stackmatrixG%*submatrixG%*subvecto rG%)sumbasisG%(swapcolG%(swaprowG%*sylvesterG%)toeplitzG%&traceG%*tran sposeG%,vandermondeG%*vecpotentG%(vectdimG%'vectorG%*wronskianG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 138 " Maple has a very strange notation for the identity matrix. It is usual ly recommended that it be \"aliased\" to a more memorable expression: \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "alias(Id=&*());" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%# IdG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 106 "Maple's input for matrices is longer winded than MATLAB's (th ere are other ways to do this; consult help)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=matrix([ [0,1],[3,2]]);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 73 "Evaluation of a matrix is not automatic. You need to tell Maple to do it;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "A;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(A);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 80 "Let's co mpute the eigenvalues of A. We start with the characteristic polynomia l:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "p:=det(A-x*Id);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p := -2*x+x^2-3;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evs:=solve(p=0);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 23 "We wa nt them separately" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ev1:= evs[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ev2:=evs[2];" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 157 "Now to get the corresponding eigenvectors we do the Gauss-Jordan \+ method, using the convenient Maple command \"backsub\". Also observe t he notation for vectors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "z:=vector([0,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A11:= augment(A-ev1*Id,z);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A12:=augment(A-ev2*Id,z);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " v1t:=backsub(gausselim(A11),false,t);" }}{PARA 11 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "v1:=subs(t[1]=1,evalm(v 1t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "v2t:=backsub(gauss elim(A12),false,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "v2:= subs(t[1]=1,evalm(v2t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 102 "Recall that the eigenvectors can be use d to diagonalize a matrix. Make them the columns of a matrix P:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "P:=evalm((augment(v1,v2))^(-1));" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 81 "Here is our diagonalizatio n. Observe the notation for the matrix inverse and the " }{TEXT 258 7 "unusual" }{TEXT -1 37 " notation for matrix multiplication. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 59 "Note the strange notation for matrix muultiplication \"&*\". " }}{PARA 0 "" 0 "" {TEXT 270 45 "But matrix inversion has \"ordinary\" notation." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Di:=evalm(P^(-1)&*A&*P);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 38 "One can also use the simpler commands:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "n ullspace(A-ev1*Id);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "null space(A-ev2*Id);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 17 "or simpler still:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "jo rdan(A,'P1');" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalm(P1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalm(P 1^(-1)&*A&*P1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 48 "FYI: Maple matrices can have var iables in them. " }}{PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 " " {TEXT -1 110 "For example, suppose I want to know the general form o f the cubic equation through (x1,y1), (x2,y2), (x3,y3). " }}{PARA 270 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 35 "First I make th e Vandermonde matrix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "V:= matrix([[1,x1,x1^2],[1,x2,x2^2],[1,x3,x3^2]]);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 31 "Then I define Y an d solve AK=Y:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Y:=vector( [y1,y2,y3]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "K:=evalm(V^(-1)&*Y)" }{TEXT -1 0 "" }}}{EXCHG {PARA 271 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 265 8 "Problem:" }{TEXT -1 68 " Finish this problem. That is, fi nd p(x) so that p(xi)=yi, i=1,2,3. " }}{PARA 273 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 267 49 "Recall that if K=[c,b,a] then P(x)=a x^2+b x +c. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 51 "Finish by integrating your p(x) from x=x1 to x=x3. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 34 "Now let's work on \+ the Jordan form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:=matrix([[3,0,0],[2,3,-1],[0,0,3]]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "p:=det(A-x*Id);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evs :=solve(p=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ev1:=evs[1 ]; ev2:=evs[2];ev3=evs[3];" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "bas:=nullspace(A-ev1*Id);" } }{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "B:=augment(A-ev1,bas[1]);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gevs:=rref(B);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "gev :=backsub(gevs,false,t);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "gev1:=subs(t[1]=0,t[2]=0,evalm(gev) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "P:=augment(bas[1],gev 1,bas[2]);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalm(P^(-1)&*A&*P);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 16 "Another ex ample:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A: =matrix([[1,2,0,0],[0,1,3,1],[0,0,1,2],[0,0,0,2]]);" }}{PARA 11 "" 1 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "p:=det(A- x*Id);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evs:=solve(p=0);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "ev1:=evs[1]; ev2:=evs[2];ev 3=evs[3];ev4=evs[4];" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "bas1:=nullspace(A-ev1*Id)[1];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "bas2:=nullspace(A-ev2*Id)[1] ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "B1:=augment(A-ev2*Id,b as2);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gevs:=rref(B1);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gev1:=backsub(gevs,false,t); " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "bas3:=subs(t[1]=0,evalm(gev));" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "B2:=augment (A-ev2*Id,bas3);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 15 "gevs:=rref(B2);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gev2:=backsub(gev s,false,t);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "bas4:=subs(t[1]=0,evalm(gev2));" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 32 "P:=augment(bas1,bas2,bas3,bas4);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " evalm(P^(-1)&*A&*P);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "56 0 0" 16 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }