{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 1 14 2 0 105 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 220 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 7 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 7 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 28 "Eigenvalues and Eigenve ctors" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "Load the linear algebra package." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with (linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 47 "1. Calc ulation of Eigenvalues and Eigenvectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Define the three matrices " } {XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "B" "6#%\" BG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "C" "6#%\"CG" }{TEXT -1 29 " a nd the 3x3 identity matrix " }{XPPEDIT 18 0 "I3" "6#%#I3G" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "A:=matrix([[2,1,1],[2,3,4] ,[-1,-1,-2]]);\nB:=matrix([[2,-1,1],[0,3,-1],[2,1,3]]);\nC:=matrix([[2 ,1,1],[2,3,2],[3,3,4]]);\nI3:=diag(1,1,1);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Calculate the characteristic polynomial of " } {XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "lambda:='lambda';\nP:=det (A-lambda*I3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "No w factor the polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 43 "Find an eigenvector for each eigenvalue of " } {XPPEDIT 18 0 "A" "6#%\"AG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lambda:=1;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "evalm(A-lambda*I3); rref(A-lambda*I3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The following Maple commands determine eigenvalu es and eigenvectors directly. Explain the output in terms of your calc ulations above." }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eigenvals(A);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvects(A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 50 "2 . Solutions of Systems of Differential Equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Describe the solutions of each of the systems" }}{PARA 0 "" 0 "" {TEXT -1 9 " " } {XPPEDIT 18 0 "dX/dt=AX" "6#/*&%#dXG\"\"\"%#dtG!\"\"%#AXG" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " " } {XPPEDIT 18 0 "dX/dt=BX" "6#/*&%#dXG\"\"\"%#dtG!\"\"%#BXG" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " " } {XPPEDIT 18 0 "dX/dt=CX" "6#/*&%#dXG\"\"\"%#dtG!\"\"%#CXG" }}{PARA 0 " " 0 "" {TEXT -1 5 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 257 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 32 "3. Eigenvalues a nd Determinants" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Find a rule relating the eigenvalues of a matrix and the \+ determinant of the matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "eigenvals(A); det(A);\neigenvals(B); det(B) ;\neigenvals(C); det(C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Check your rule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "E:=randmatrix(3,3,symmetric);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalf(Eigenvals(E));det(E);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 45 "Give an algebraic justification of your r ule." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 22 "4. The Trace Operation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Examine t he calculations below, and formulate a rule for the calculation of th e trace of a matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "evalm(A);trace(A);\nevalm(B);trace(B);\nevalm(C) ;trace(C);\nF:=diag(2,4,3);trace(F);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 42 "Check your rule for calculating the trace." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "G:= randmatrix(3,3);trace(G);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "Examine the calculations below, and formulate a rule relating the trace of a matrix to the eigenvalues of the matrix." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "eigenvals(A);trac e(A);\neigenvals(B);trace(B);\neigenvals(C);trace(C);\neigenvals(F);tr ace(F);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "Using bo th symmetric and non-symmetric matrices, check your rule connecting th e trace of a matrix to the eigenvalues of the matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "H:=randmatrix(3, 3,symmetric);\nevalf(Eigenvals(H)); trace(H);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 77 "Explain why your rule must be true for th e special case of diagonal matrices." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 11 "5. Summary" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 4 0" 14 }{VIEWOPTS 1 1 0 1 1 1803 }