{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 5 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 259 24 "Linear A lgebra Powertool" }}{PARA 256 "" 0 "" {TEXT -1 25 "Creating Example Ma trices" }}{PARA 0 "" 0 "" {TEXT 260 26 "Worksheet by Russell Blyth" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 288 "The material in \+ chapter 7 on canonical forms is rather dense. It is useful to look at examples with the material. A minor difficulty is that any computati on on 8 by 8 matrices gets tedious in a hurry. That is where Maple co mes in. We start by loading Maple's linear algebra functions." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}}{EXCHG {PARA 5 "" 0 "" {TEXT 258 22 "Creating a Jordan bloc" }{TEXT -1 0 "" } {TEXT 256 0 "" }{TEXT -1 0 "" }{TEXT 257 9 "k matrix." }{TEXT -1 1 " \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 187 "The first kind of matrix we \+ would like to create is a specified Jordan block. To do this we use t he JordanBock command. Consider the Jordan block of size 4 associated with eigenvalue 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "J2s4 := JordanBlock(2,4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "Next we would like to put several Jordan blocks together in a single matrix. \+ We do that with the diag command. Consider the matrix used in exampl e 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "Examp1 := diag(Jord anBlock(2,3),JordanBlock(2,1),JordanBlock(3,2), JordanBlock(0,2));" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "We also want to be able to produc e an identity matrix of the same size." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Ident8 := diag(seq(1,i=1..8));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "We can now look at matrices of the form A-lambda*I a nd its powers." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "evalm(Exa mp1-2*Ident8);\nevalm((Examp1-2*Ident8)^2);\nevalm((Examp1-2*Ident8)^3 );\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "Notice that the Jordan b locks associated with 2 are killed by sufficiently high powers of (M-2 I) and other blocks keep the same size." }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 22 "Finding a Jordan basis" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "To find a Jordan basis for M, we need to look at the nulspaces of (M-lambda*I)^i for various lambda and i." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 273 "We continue to work with the same example. (You shoul d stop and not that since the matrix is in Jordan form, the canonical \+ basis is a Jordan basis. Nevertheless, it is a good example for worki ng on technique.) We want to look at the nullspces of (M-2I) ^n for v arious n." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "Mat1 := evalm (Examp1-2*Ident8);\nans1 := nullspace(Mat1);\nans2 := nullspace(evalm( Mat1^2));\nans3 := nullspace(evalm(Mat1^3));\nans4 := nullspace(evalm( Mat1^4));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 193 "Time for some techn ical tidying. Maple answers the nullspace command with a set of basis vectors for the nullspace. Sets are unordered. To turn the sets int o sequences we use the op command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "nul1 := op(ans1);\nnul2 := op(ans2);\nnul3 := op(ans3 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 254 "Looking at the sizes of th e nullspaces, we need a 3-cycle that starts with a vector in nul3 that is not in the span of nul2, and a 1-cycle that starts with a vector i n nul1 that is linearly independent form the set of initial vectors of the longer cycle." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "To check l inear independence of the vectors in nul3 from the vectors in nul2 I w ant to turn the vectors in nul2 into a matrix with the stackmatrix com mand." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "test1 := stackmatr ix(nul2[1], nul2[2], nul2[3]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Next I add each vector in nul3 to that matrix and see if the rank inc reases." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "for i from 1 to \+ 4 do\nrank(stackmatrix(test1,nul3[i]));\nod;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 36 "So we see that v1 should be nul3[2]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v1 := nul3[2];" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 44 "Next we find the other vectors in the cycle." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "v1a := evalm(Mat1&*v1);\nv1b := evalm(Mat1&*v1a);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Now we l ook for vectors in nul1 that are LI from v1b." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "for i from 1 to 2 do\nrank(stackmatrix(v1b, nul1 [i]));\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v2 := nul1[1 ];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Thus the Jordan basis for K _2 is \{v1b, v1a, v1, v2\}." }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 8 "Exe rcise" }}{PARA 0 "" 0 "" {TEXT -1 75 "Find the Jordan basis for the fo llowing matrix which is not in Jordan form." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "M2 := diag(companion((x-3)^4, x), companion((x-3) ^2,x),\n companion((x-3)^2,x), companion(x-3,x));\nfactor(charpoly( M2,x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 2 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }