{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 }{PSTYLE "N ormal" -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 "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }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 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 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 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 31 "Introduction to Eigenvalues and " }}{PARA 18 "" 0 "" {TEXT -1 12 "Eigenvectors" }}{PARA 19 "" 0 "" {TEXT -1 22 "Math 374 Fall 1999" }}{PARA 19 "" 0 "" {TEXT -1 11 "J ay Treiman" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Load the " }{HYPERLNK 17 "linalg" 2 "linalg" "" }{TEXT -1 9 " package." }}{PARA 0 "" 0 "" {TEXT -1 50 "It contains the routines \+ for working with matices." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;\nwith(linalg):" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "The definition of an eigenpair." }}{PARA 0 "" 0 "" {TEXT -1 39 "Recall th at an eigenvalue for a matrix " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 13 " is a number " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 10 " such that" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "A*v = lambda*v;" "6#/*&%\"AG\"\"\"%\"vGF&*&%'lambdaGF&F'F&" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "for a nonzero vector " } {XPPEDIT 18 0 "v;" "6#%\"vG" }{TEXT -1 52 ". An eigenvalue may have m ore than one eigenvector." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examp les" }}{PARA 0 "" 0 "" {TEXT -1 58 "Here are some simple examples. Fi rst is 2x2 a matrix with" }}{PARA 0 "" 0 "" {TEXT -1 40 "two distinct \+ eigenvalues. Note how the " }{HYPERLNK 17 "eigenvects" 2 "linalg,eige nvects" "" }{TEXT -1 8 " command" }}{PARA 0 "" 0 "" {TEXT -1 44 "calcu lates the eigenvalues and eigenvectors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "A := matrix([[2,1], [1, 2]]);\neigenvects(A);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Here is an example with only two eigenvalues but " } }{PARA 0 "" 0 "" {TEXT -1 32 "3 eigenvectors for a 3x3 matrix." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "B := matrix([[2,1,0],[1,2,0],[0,0,1]]);\neigenvects(B);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Next comes an e xample with only one eigenvalue and only" }}{PARA 0 "" 0 "" {TEXT -1 33 "one eigenvector for a 2x2 matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "C := matrix([[1,-1],[0,1 ]]);\neigenvects(C);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Finally is an example with only complex eigenvalues" } }{PARA 0 "" 0 "" {TEXT -1 25 "and complex eigenvectors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "H := ([ [1,2],[-2,1]]);\neigenvects(H);" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "Finding eigenvalues by hand." }}{PARA 0 "" 0 "" {TEXT -1 60 "Th e method for finding eigenvalues by hand is fairly simple." }}{PARA 0 "" 0 "" {TEXT -1 20 "The matrix equation " }{XPPEDIT 18 0 "A*v = lambd a*v;" "6#/*&%\"AG\"\"\"%\"vGF&*&%'lambdaGF&F'F&" }{TEXT -1 23 " has a \+ nonzero solution" }}{PARA 0 "" 0 "" {TEXT -1 35 "if and only if the ma trix equation " }{XPPEDIT 18 0 "(A-lambda*I)*v = 0;" "6#/*&,&%\"AG\"\" \"*&%'lambdaGF'%\"IGF'!\"\"F'%\"vGF'\"\"!" }{TEXT -1 6 " has a" }} {PARA 0 "" 0 "" {TEXT -1 47 "nonzero solution. This happens if and on ly if " }{XPPEDIT 18 0 "det(A-lambda*I) = 0;" "6#/-%$detG6#,&%\"AG\"\" \"*&%'lambdaGF)%\"IGF)!\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 " Because of this only tr ies to find the roots of the characteristic" }}{PARA 0 "" 0 "" {TEXT -1 14 "polynomial of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "det(A-lambda*I);" "6#-%$detG6#,&%\"AG\"\"\"*&%'lambdaG F(%\"IGF(!\"\"" }{TEXT -1 31 ". This can be a very hard job," }} {PARA 0 "" 0 "" {TEXT -1 71 "but is about the best one can do by hand. (For large systems one uses " }}{PARA 0 "" 0 "" {TEXT -1 59 "differe nt methods that we will not consider in this class.)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Consider the matrix " } {XPPEDIT 18 0 "A = matrix([[3, 1], [1, 3]]);" "6#/%\"AG-%'matrixG6#7$7 $\"\"$\"\"\"7$\"\"\"\"\"$" }{TEXT -1 32 ". The characteristic polynom ial" }}{PARA 0 "" 0 "" {TEXT -1 57 "is easily calculated one one defin es the identity matrix." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "J := matrix([[3,1],[1,3]]);\nident := diag(1,1);\nmat_1 := evalm(J-lambda*ident);\nchar_poly := det(mat_ 1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "O ne can then find the roots of the characteristic polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ei gen_vals := solve(char_poly,lambda);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 65 "This has two distinct eigenvalues. O ne can get the corresponding" }}{PARA 0 "" 0 "" {TEXT -1 23 "eigenvect ors using the " }{HYPERLNK 17 "nullspace" 2 "linalg,nullspace" "" } {TEXT -1 34 " command to find nonzero solutions" }}{PARA 0 "" 0 "" {TEXT -1 4 "of " }{XPPEDIT 18 0 "A*v = b;" "6#/*&%\"AG\"\"\"%\"vGF&% \"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "b:=vector([0,0]);\nJ1 := evalm(J-eigen_v als[1]*ident);\nv1 := nullspace(J1);\nJ2 := evalm(J-eigen_vals[2]*iden t);\nv2 := nullspace(J2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Exercises " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Check that Maple has given eig envalues and eigenvectors" }}{PARA 0 "" 0 "" {TEXT -1 29 "for the matr ices A, B, and C." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Check that t he functions " }{XPPEDIT 18 0 "v[i]*exp(lambda[i]*t);" "6#*&&%\"vG6#% \"iG\"\"\"-%$expG6#*&&%'lambdaG6#F'F(%\"tGF(F(" }{TEXT -1 17 " are sol utions to" }}{PARA 0 "" 0 "" {TEXT -1 26 "the differential equation " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(y,t) = B*y;" " 6#/-%%diffG6$%\"yG%\"tG*&%\"BG\"\"\"F'F+" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "for i = 1, 2, and 3." }}}}}{MARK "9" 0 }{VIEWOPTS 1 1 0 1 1 1803 }