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} {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 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 {PARA 3 "" 0 "" {TEXT -1 29 "Eigenvalues and Eigenvectors " }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 74 "Systems of equations with infinitely many solutions and singular matrices " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 34 "Consider the system of equations: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2*x+4*y+z \+ = 0, ``],[x+3*y+2*z = 0, ``],[x-3*y-7*z = 0, ``]);" "6#-%*PIECEWISEG6% 7$/,(*&\"\"#\"\"\"%\"xGF+F+*&\"\"%F+%\"yGF+F+%\"zGF+\"\"!%!G7$/,(F,F+* &\"\"$F+F/F+F+*&F*F+F0F+F+F1F27$/,(F,F+*&F7F+F/F+!\"\"*&\"\"(F+F0F+F=F 1F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 35 "The equivalent ma trix equation is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[2, 4, 1], [1, 3, 2], [1, -3, -7]])*matrix([[x], [y], [z]]) = matrix([[0], [0], [0]]);" "6#/*&-%'matrixG6#7%7%\"\"#\"\"%\"\"\"7%F, \"\"$F*7%F,,$F.!\"\",$\"\"(F1F,-F&6#7%7#%\"xG7#%\"yG7#%\"zGF,-F&6#7%7# \"\"!7#FA7#FA" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "The sys tem can be solved by Gaussian-Jordan elimination. " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[2, 4, 1, `|`, 0], [1, 3, 2, \+ `|`, 0], [1, -3, -7, `|`, 0]]);" "6#-%'matrixG6#7%7'\"\"#\"\"%\"\"\"% \"|grG\"\"!7'F*\"\"$F(F+F,7'F*,$F.!\"\",$\"\"(F1F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 7 " swap " }{XPPEDIT 18 0 "R[1]" "6#&%\"RG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "R[2]" "6#&%\"RG6#\"\"#" }{TEXT -1 7 " ... " } {XPPEDIT 18 0 "matrix([[1, 3, 2, `|`, 0], [2, 4, 1, `|`, 0], [1, -3, - 7, `|`, 0]]);" "6#-%'matrixG6#7%7'\"\"\"\"\"$\"\"#%\"|grG\"\"!7'F*\"\" %F(F+F,7'F(,$F)!\"\",$\"\"(F1F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECE WISE([R[2] = R[2]-2*R[1], ``],[R[3] = R[3]-R[1], ``]);" "6#-%*PIECEWIS EG6$7$/&%\"RG6#\"\"#,&&F)6#F+\"\"\"*&F+F/&F)6#F/F/!\"\"%!G7$/&F)6#\"\" $,&&F)6#F9F/&F)6#F/F3F4" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix( [[1, 3, 2, `|`, 0], [0, -2, -3, `|`, 0], [0, -6, -9, `|`, 0]]);" "6#-% 'matrixG6#7%7'\"\"\"\"\"$\"\"#%\"|grG\"\"!7'F,,$F*!\"\",$F)F/F+F,7'F,, $\"\"'F/,$\"\"*F/F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R[2] = -R[2], ``],[R[3] = -R[3], ``])" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"#,$& F)6#F+!\"\"%!G7$/&F)6#\"\"$,$&F)6#F5F/F0" }{TEXT -1 7 " ... " } {XPPEDIT 18 0 "matrix([[1, 3, 2, `|`, 0], [0, 2, 3, `|`, 0], [0, 6, 9, `|`, 0]])" "6#-%'matrixG6#7%7'\"\"\"\"\"$\"\"#%\"|grG\"\"!7'F,F*F)F+F ,7'F,\"\"'\"\"*F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[3]=R[3]-3*R[2]" "6 #/&%\"RG6#\"\"$,&&F%6#F'\"\"\"*&F'F+&F%6#\"\"#F+!\"\"" }{TEXT -1 7 " \+ ... " }{XPPEDIT 18 0 "matrix([[1, 3, 2, `|`, 0], [0, 2, 3, `|`, 0], [ 0, 0, 0, `|`, 0]])" "6#-%'matrixG6#7%7'\"\"\"\"\"$\"\"#%\"|grG\"\"!7'F ,F*F)F+F,7'F,F,F,F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[2] = R[2]*`.`*` `(1/2);" "6#/&%\"RG6#\"\"#*(&F%6#F'\"\"\"%\".GF+-%!G6#*&F+F+F'!\"\"F+ " }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[1, 3, 2, `|`, 0], [0, 1, 3/2, `|`, 0], [0, 0, 0, `|`, 0]]);" "6#-%'matrixG6#7%7'\"\"\"\"\"$ \"\"#%\"|grG\"\"!7'F,F(*&F)F(F*!\"\"F+F,7'F,F,F,F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R[1] = R[1]-3*R[2];" "6#/&%\"RG6#\"\"\",&&F%6#F'F'*&\" \"$F'&F%6#\"\"#F'!\"\"" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([ [1, 0, -5/2, `|`, 0], [0, 1, 3/2, `|`, 0], [0, 0, 0, `|`, 0]]);" "6#-% 'matrixG6#7%7'\"\"\"\"\"!,$*&\"\"&F(\"\"#!\"\"F.%\"|grGF)7'F)F(*&\"\"$ F(F-F.F/F)7'F)F)F)F/F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "PIECEWISE([ x-5/2*z=0,``],[y+3/2*z=0,``])" "6#-%*PIECEW ISEG6$7$/,&%\"xG\"\"\"*(\"\"&F*\"\"#!\"\"%\"zGF*F.\"\"!%!G7$/,&%\"yGF* *(\"\"$F*F-F.F/F*F*F0F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "z=2*t" "6#/%\"zG*&\"\"#\"\"\"%\"tGF'" }{TEXT -1 7 ". Then " }{XPPEDIT 18 0 "x=5*t" "6#/%\"xG*&\"\"&\"\"\"%\"tGF'" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y=-3*t" "6#/%\"yG,$*&\"\"$\"\"\"%\" tGF(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "There are i nfinitely many solutions given by the parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x=5*t,``],[y=-3* t,``],[z=2*t,``])" "6#-%*PIECEWISEG6%7$/%\"xG*&\"\"&\"\"\"%\"tGF+%!G7$ /%\"yG,$*&\"\"$F+F,F+!\"\"F-7$/%\"zG*&\"\"#F+F,F+F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 72 "These equations can be written in the f orm of a single vector equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x], [y], [z]])=t*matrix([[5], [-3], [2]])" "6 #/-%'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG*&%\"tG\"\"\"-F%6#7%7#\"\"&7#,$\" \"$!\"\"7#\"\"#F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 93 "whic h represents a line through the origin in 3-dimensional space with the direction vector: " }}{PARA 256 "" 0 "" {TEXT -1 3 " 5 " }{TEXT 263 1 "i" }{XPPEDIT 18 0 "``-3" "6#,&%!G\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 264 1 "j" }{XPPEDIT 18 0 "``+2" "6#,&%!G\"\"\"\"\"#F%" }{TEXT -1 1 " " }{TEXT 265 1 "k" }{XPPEDIT 18 0 "`` = matrix([[5],[-3],[2]]) " "6#/%!G-%'matrixG6#7%7#\"\"&7#,$\"\"$!\"\"7#\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Here " }{TEXT 266 1 "i" }{XPPEDIT 18 0 " ``=matrix([[1], [0], [0]])" "6#/%!G-%'matrixG6#7%7#\"\"\"7#\"\"!7#F," }{TEXT -1 2 ", " }{TEXT 267 1 "j" }{XPPEDIT 18 0 "``=matrix([[0], [1], [0]])" "6#/%!G-%'matrixG6#7%7#\"\"!7#\"\"\"7#F*" }{TEXT -1 5 " and " }{TEXT 268 1 "k" }{XPPEDIT 18 0 "`` = matrix([[0], [0], [1]])" "6#/%!G -%'matrixG6#7%7#\"\"!7#F*7#\"\"\"" }{TEXT -1 62 " are the standard bas is vectors in the coordinate directions. " }}{PARA 0 "" 0 "" {TEXT -1 44 "In particular, we can check that the values " }{XPPEDIT 18 0 "x=5 " "6#/%\"xG\"\"&" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=-3" "6#/%\"yG,$\" \"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "z=2" "6#/%\"zG\"\"#" } {TEXT -1 60 " provide a solution by the following matrix multiplicatio n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "matrix([[2, 4, 1], [1, 3, 2], [1, -3, -7]])*matrix([ [5], [-3], [2]]) = matrix([[0], [0], [0]]);" "6#/*&-%'matrixG6#7%7%\" \"#\"\"%\"\"\"7%F,\"\"$F*7%F,,$F.!\"\",$\"\"(F1F,-F&6#7%7#\"\"&7#,$F.F 17#F*F,-F&6#7%7#\"\"!7#F@7#F@" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 259 21 "Important observation" }{TEXT -1 2 ": " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "The matrix " }{XPPEDIT 18 0 "A=matrix([[2, 4, 1], [1, 3, 2], [1, -3, -7]])" "6#/%\"AG-%'matri xG6#7%7%\"\"#\"\"%\"\"\"7%F,\"\"$F*7%F,,$F.!\"\",$\"\"(F1" }{TEXT -1 4 " is " }{TEXT 259 8 "singular" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "det(A)=0" "6#/-%$detG6#%\"AG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 32 "This is necessarily so since if " }{XPPEDIT 18 0 " det(A)" "6#-%$detG6#%\"AG" }{TEXT -1 78 " is not 0 then this would imp ly that the only solution of the matrix equation " }{XPPEDIT 18 0 "A*` .`*X=``" "6#/*(%\"AG\"\"\"%\".GF&%\"XGF&%!G" }{TEXT 269 1 "0" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "X=matrix([[x], [y], [z]])" "6#/%\"XG-% 'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG" }{TEXT -1 5 " and " }{TEXT 270 1 "0 " }{XPPEDIT 18 0 "``=matrix([[0], [0], [0]])" "6#/%!G-%'matrixG6#7%7# \"\"!7#F*7#F*" }{TEXT -1 26 ", is the trivial solution " }{XPPEDIT 18 0 "X=matrix([[0], [0], [0]])" "6#/%\"XG-%'matrixG6#7%7#\"\"!7#F*7#F*" }{TEXT -1 9 ". Indeed " }{XPPEDIT 18 0 "det(A)<>0" "6#0-%$detG6#%\"AG \"\"!" }{TEXT -1 14 " implies that " }{TEXT 271 1 "A" }{TEXT -1 16 " h as an inverse " }{XPPEDIT 18 0 "A^(-1)" "6#)%\"AG,$\"\"\"!\"\"" } {TEXT -1 10 " so that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "X=I*`.`*X" "6#/%\"XG*(%\"IG\"\"\"%\".GF'F$F'" }{XPPEDIT 18 0 "`` =A^(-1)*`.`*A*`.`*X" "6#/%!G*,)%\"AG,$\"\"\"!\"\"F)%\".GF)F'F)F+F)%\"X GF)" }{XPPEDIT 18 0 "``=A^(-1)*`.`" "6#/%!G*&)%\"AG,$\"\"\"!\"\"F)%\". GF)" }{TEXT -1 1 " " }{TEXT 272 1 "0" }{XPPEDIT 18 0 "``=``" "6#/%!GF$ " }{TEXT 273 1 "0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 77 "Thi s contradicts the fact that there are non-zero solutions, so we must h ave " }{XPPEDIT 18 0 "det(A)=0" "6#/-%$detG6#%\"AG\"\"!" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 23 "It is easy to evaluate " }{XPPEDIT 18 0 "det(A)" "6#-%$detG6#%\"AG" }{TEXT -1 25 " and check that it is 0 . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 279 192 192 {PLOTDATA 2 "6B-%'CURVESG6$7$7$$!3A+++++++S!#=$!35+++++++?F*7$F($\"33+ ++++++@!#<-%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$\"3!**************R#F0F +7$F:F.F1-F$6(7$7$$!\"\"!\"#FA7$$\"$q#FCFE7%7$$\"+\"*pY6D!\"*$\"+4I`FCFA7$$\"$q%FCFE7 %7$$\"+\"*pY6XFKFLF_p7$$\"+4I`9&FKF]q7$$\"+'*Rk6DFK$!*C%41iFKFO-FT6&F4FX$\" \"$FBF^rFYFgn-F$6(7$7$F]oF]p7$F`oF^q7%7$$\"+/gN " 0 "" {MPLTEXT 1 0 53 "solve( \{2*x+4*y+z=0,x+3*y+2*z=0,x-3*y-7*z=0\},\{x,y,z\});" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<%/%\"yG,$*(\"\"$\"\"\"\"\"#!\"\"%\"zGF)F+/%\"xG,$*( \"\"&F)F*F+F,F)F)/F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Eigenvalues and eigenvectors " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 307 1 "A" }{TEXT -1 19 " be a square matrix" }{TEXT 274 2 ". " }{TEXT -1 48 " Suppose that there is a non-zero column vector " }{TEXT 275 1 "X " }{TEXT -1 19 " and a real number " }{XPPEDIT 18 0 "lambda" "6#%'lamb daG" }{TEXT -1 11 " such that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "A*`.`*X=lambda*`.`*X" "6#/*(%\"AG\"\"\"%\".GF&%\"XGF&*( %'lambdaGF&F'F&F(F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Th en " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 14 " is called an " }{TEXT 259 10 "eigenvalue" }{TEXT -1 4 " of " }{TEXT 276 1 "A" } {TEXT -1 31 ", and the corresponding vector " }{TEXT 277 1 "X" }{TEXT -1 14 " is called an " }{TEXT 259 11 "eigenvector" }{TEXT -1 4 " of " }{TEXT 278 1 "A" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "In gen eral an " }{TEXT 281 1 "n" }{TEXT -1 1 " " }{TEXT 279 1 "x" }{TEXT -1 1 " " }{TEXT 280 1 "n" }{TEXT -1 15 " square matrix " }{TEXT 282 1 "A " }{TEXT -1 16 " may have up to " }{TEXT 283 1 "n" }{TEXT -1 83 " eige nvalues, which are solutions of a polynomial equation determined as fo llows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A*`.`*X=lambda*`.`*X" "6#/*(%\"AG\"\"\"%\".GF &%\"XGF&*(%'lambdaGF&F'F&F(F&" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 " X <>``" "6#0%\"XG%!G" }{TEXT 286 1 "0" }{TEXT -1 3 ", " }}{PARA 256 " " 0 "" {TEXT -1 19 " if and only if " }{XPPEDIT 18 0 "A*`.`*X - lam bda*`.`*X=``" "6#/,&*(%\"AG\"\"\"%\".GF'%\"XGF'F'*(%'lambdaGF'F(F'F)F' !\"\"%!G" }{TEXT 284 1 "0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 19 " if and only if " }{XPPEDIT 18 0 "(A-lambda*`.`*I)*`.`*X = ` `;" "6#/*(,&%\"AG\"\"\"*(%'lambdaGF'%\".GF'%\"IGF'!\"\"F'F*F'%\"XGF'%! G" }{TEXT 285 1 "0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 18 " \+ if and only 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 1 " " }{XPPEDIT 18 0 "det(A-lambda*I) = 0" "6#/-%$detG6#,&%\"AG\"\"\"*&%'lambdaGF)%\"IGF) !\"\"\"\"!" }{TEXT -1 63 " is the required polynomial equation which h as the eigenvalues " }{XPPEDIT 18 0 "lambda=lambda[1], lambda[2], ` . \+ . . `" "6%/%'lambdaG&F$6#\"\"\"&F$6#\"\"#%(~.~.~.~G" }{TEXT -1 19 " as its solutions. " }}{PARA 0 "" 0 "" {TEXT -1 39 "This polynomial equat ion is called the " }{TEXT 259 23 "characteristic equation" }{TEXT -1 4 " of " }{TEXT 287 1 "A" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The problem of finding the eigenva lues and eigenvectors of a square matrix " }{TEXT 289 1 "A" }{TEXT -1 40 " involves first finding the eigenvalues " }{XPPEDIT 18 0 "lambda[1 ],lambda[2],` . . . `" "6%&%'lambdaG6#\"\"\"&F$6#\"\"#%(~.~.~.~G" } {TEXT -1 49 " as the solutions of the characteristic equation " } {XPPEDIT 18 0 "det(A-lambda*I) = 0" "6#/-%$detG6#,&%\"AG\"\"\"*&%'lamb daGF)%\"IGF)!\"\"\"\"!" }{TEXT -1 75 ". Corresponding eigenvectors are then found by solving the matrix equation " }{XPPEDIT 18 0 "(A-lambda [k]*`.`*I)*`.`*X = ``;" "6#/*(,&%\"AG\"\"\"*(&%'lambdaG6#%\"kGF'%\".GF '%\"IGF'!\"\"F'F-F'%\"XGF'%!G" }{TEXT 288 1 "0" }{TEXT -1 31 " for ea ch specific eigenvalue " }{XPPEDIT 18 0 "lambda[k]" "6#&%'lambdaG6#%\" kG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "E xamples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT 290 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 52 "Find the \+ eigenvalues and eigenvectors of the matrix " }{XPPEDIT 18 0 "A=matrix( [[3,2,-2],[-1,-4,1],[2,-4,-1]])" "6#/%\"AG-%'matrixG6#7%7%\"\"$\"\"#,$ F+!\"\"7%,$\"\"\"F-,$\"\"%F-F07%F+,$F2F-,$F0F-" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT 291 8 "Solution" }{TEXT -1 2 ": " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A-lambda*I = matrix([[3, 2, -2], [-1, -4, 1], [2, -4, -1]])-lambda*matrix([[1, 0, 0], [0, 1, 0], [0, 0 , 1]]);" "6#/,&%\"AG\"\"\"*&%'lambdaGF&%\"IGF&!\"\",&-%'matrixG6#7%7% \"\"$\"\"#,$F2F*7%,$F&F*,$\"\"%F*F&7%F2,$F7F*,$F&F*F&*&F(F&-F-6#7%7%F& \"\"!F@7%F@F&F@7%F@F@F&F&F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=matrix([ [3-lambda, 2, -2], [-1, -4-lambda, 1], [2, -4, -1-lambda]])" "6#/%!G-% 'matrixG6#7%7%,&\"\"$\"\"\"%'lambdaG!\"\"\"\"#,$F/F.7%,$F,F.,&\"\"%F.F -F.F,7%F/,$F4F.,&F,F.F-F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "The characteristic equation of " }{TEXT 292 1 "A" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 224 135 135 {PLOTDATA 2 "61-%'CURVESG6$7$7$$!3A+++++++S!#=$!35+++++++?F*7$F($\"33+ ++++++@!#<-%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$\"3!**************R#F0F +7$F:F.F1-%%TEXTG6&7$$F5F5$\"\"#F5Q$3-l6\"F1-%%FONTG6$%'SYMBOLG\"#7-F> 6&7$$\"\"\"F5FBQ\"2FEF1FF-F>6&7$FBFBQ#-2FEF1FF-F>6&7$FAFNQ#-1FEF1FF-F> 6&7$FNFNQ%-4-lFEF1FF-F>6&7$FBFNQ\"1FEF1FF-F>6&7$FAFAFPF1FF-F>6&7$FNFAQ #-4FEF1FF-F>6&7$FBFAQ%-1-lFEF1FF-F>6%7$$\"#G!\"\"FNQ%=~0.FE-FG6%%&TIME SG%&ROMANGFJ-%+AXESLABELSG6$Q!FEFdp-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$%(D EFAULTGF\\q" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 " Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" " Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" }}{TEXT -1 3 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 292 177 177 {PLOTDATA 2 "6<-%'CURVESG6$7$7$$!3A+++++++S!#=$!35+++++++?F*7$F($\"33+++++++@!#< -%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$\"3!**************R#F0F+7$F:F.F1- F$6(7$7$$!\"\"!\"#FA7$$\"$I#FCFE7%7$$\"+$ft?9#!\"*$\"+2k#pA#FKFD7$FLFI -%&STYLEG6#%,PATCHNOGRIDG-%&COLORG6&F4$\"\"'FBFV\"\"\"-%*THICKNESSG6# \"\"#-%*LINESTYLEGFen-F$6(7$7$$\"#**FCFA7$$\"$I$FCFE7%7$$\"+$ft?9$FKFL F_o7$$\"+2k#pA$FKFIFOFSFYFgn-F$6(7$7$$\"$*>FCFA7$$\"$I%FCFE7%7$$\"+\"* pYJTFK$\"+4I`PAFKF_p7$$\"+4I`PUFK$\"+\"*pYJ@FKFOFSFYFgn-F$6(7$7$FAF]p7 $FE$!$-$!\"$7%7$$\"+u^KPAFK$!*o*fT8FKFaq7$$\"+E[nJ@FK$!*K+kS#FKFO-FT6& F4FX$\"\"$FBFbrFYFgn-F$6(7$7$F]oF]p7$F`oFbq7%7$$\"+u^KPKFKFiqFhr7$$\"+ E[nJJFKF^rFOF`rFYFgn-F$6(7$7$F]pF]p7$F`pFbq7%7$$\"+u^KPUFKFiqFds7$$\"+ E[nJTFKF^rFOF`rFYFgn-%%TEXTG6&7$$F5F5$FfnF5Q$3-l6\"F1-%%FONTG6$%'SYMBO LG\"#7-F]t6&7$$FXF5FatQ\"2FctF1Fdt-F]t6&7$FatFatQ#-2FctF1Fdt-F]t6&7$F` tF\\uQ#-1FctF1Fdt-F]t6&7$F\\uF\\uQ%-4-lFctF1Fdt-F]t6&7$FatF\\uQ\"1FctF 1Fdt-F]t6&7$F`tF`tF]uF1Fdt-F]t6&7$F\\uF`tQ#-4FctF1Fdt-F]t6&7$FatF`tQ%- 1-lFctF1Fdt-F]t6&7$$FcrF5FatFbt-FT6&F4F5FVF5Fdt-F]t6&7$$\"\"%F5FatF]uF ]wFdt-F]t6&7$F\\wF\\uFeuF]wFdt-F]t6&7$FbwF\\uFiuF]wFdt-F]t6&7$F\\wF`tF ]uF]wFdt-F]t6&7$FbwF`tFdvF]wFdt-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q !FctFgx-%%VIEWG6$%(DEFAULTGF[y" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 58 "The characteristic equation can b e simplified as follows. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(3-lambda)*(-4-lambda)*(-1-lambda)+4-8-``(16+4*lambda-1 2+4*lambda+2+2*lambda)=0" "6#/,**(,&\"\"$\"\"\"%'lambdaG!\"\"F(,&\"\"% F*F)F*F(,&F(F*F)F*F(F(F,F(\"\")F*-%!G6#,.\"#;F(*&F,F(F)F(F(\"#7F**&F,F (F)F(F(\"\"#F(*&F7F(F)F(F(F*\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "(3-lambda)*(4+5*lambda+lambda^2)-4-(6 +10*lambda) = 0;" "6#/,(*&,&\"\"$\"\"\"%'lambdaG!\"\"F(,(\"\"%F(*&\"\" &F(F)F(F(*$F)\"\"#F(F(F(F,F*,&\"\"'F(*&\"#5F(F)F(F(F*\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "12+15*lambda+ 3*lambda^2-4*lambda-5*lambda^2-lambda^3-10-10*lambda = 0;" "6#/,2\"#7 \"\"\"*&\"#:F&%'lambdaGF&F&*&\"\"$F&*$F)\"\"#F&F&*&\"\"%F&F)F&!\"\"*& \"\"&F&*$F)F-F&F0*$F)F+F0\"#5F0*&F5F&F)F&F0\"\"!" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-lambda^3-2*lambda^2+ lambda+2=0" "6#/,**$%'lambdaG\"\"$!\"\"*&\"\"#\"\"\"*$F&F*F+F(F&F+F*F+ \"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda^3+2*lambda^2-lambda-2 = 0;" "6#/,**$%'lambdaG\"\"$\"\"\"* &\"\"#F(*$F&F*F(F(F&!\"\"F*F,\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda^2*(lambda+2)-(lambda+2)=0" "6# /,&*&%'lambdaG\"\"#,&F&\"\"\"F'F)F)F),&F&F)F'F)!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(lambda^2-1)* (lambda+2)=0" "6#/*&,&*$%'lambdaG\"\"#\"\"\"F)!\"\"F),&F'F)F(F)F)\"\"! " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "( lambda-1)*(lambda+1)*(lambda+2) = 0;" "6#/*(,&%'lambdaG\"\"\"F'!\"\"F' ,&F&F'F'F'F',&F&F'\"\"#F'F'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "The eigenvalues are " }{XPPEDIT 18 0 "lambda=1" "6#/%'lam bdaG\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "lambda=-1" "6#/%'lambdaG, $\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "lambda = -2;" "6#/%' lambdaG,$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 44 "Eigenvectors associated with the eigen value " }{XPPEDIT 18 0 "lambda = -2;" "6#/%'lambdaG,$\"\"#!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "Consider the equation \+ " }{XPPEDIT 18 0 "(A+2*I)*`.`*X = ``;" "6#/*(,&%\"AG\"\"\"*&\"\"#F'%\" IGF'F'F'%\".GF'%\"XGF'%!G" }{TEXT 293 1 "0" }{TEXT -1 11 ", that is, \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[5, 2, -2 ], [-1, -2, 1], [2, -4, 1]])*matrix([[x], [y], [z]]) = matrix([[0], [0 ], [0]]);" "6#/*&-%'matrixG6#7%7%\"\"&\"\"#,$F+!\"\"7%,$\"\"\"F-,$F+F- F07%F+,$\"\"%F-F0F0-F&6#7%7#%\"xG7#%\"yG7#%\"zGF0-F&6#7%7#\"\"!7#FB7#F B" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "X=matrix([[x],[y],[z]])" "6# /%\"XG-%'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 143 "It is not neces sary to augment the coefficient matrix with a column of zeros because \+ such a column remains unchanged under all row operations. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[5, 2, -2], [-1, -2, 1], [2, -4, 1]])" "6#-%'matrixG6#7%7% \"\"&\"\"#,$F)!\"\"7%,$\"\"\"F+,$F)F+F.7%F),$\"\"%F+F." }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R[2]=-R[2]" "6#/&%\"RG6#\"\"#,$&F%6#F'!\"\"" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[5, 2, -2], [1, 2, -1], [2, -4, 1] ])" "6#-%'matrixG6#7%7%\"\"&\"\"#,$F)!\"\"7%\"\"\"F),$F-F+7%F),$\"\"%F +F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 7 " swap " }{XPPEDIT 18 0 "R[1]" "6#&%\"RG6#\"\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "R[2]" "6#&%\"RG6#\"\"#" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([ [1, 2, -1],[5, 2, -2], [2, -4, 1] ])" "6#-%'matrixG6#7%7%\"\"\"\"\"#,$F(!\"\"7%\"\"&F),$F)F+7%F),$\"\"%F +F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R[2]=R[2]-5*R[1],``],[R[3 ]=R[3]-2*R[1],``])" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"#,&&F)6#F+\"\"\"* &\"\"&F/&F)6#F/F/!\"\"%!G7$/&F)6#\"\"$,&&F)6#F:F/*&F+F/&F)6#F/F/F4F5" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([ [1, 2, -1],[0, -8, 3], \+ [0, -8, 3]])" "6#-%'matrixG6#7%7%\"\"\"\"\"#,$F(!\"\"7%\"\"!,$\"\")F+ \"\"$7%F-,$F/F+F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[3]=R[3]-R[2]" "6#/& %\"RG6#\"\"$,&&F%6#F'\"\"\"&F%6#\"\"#!\"\"" }{TEXT -1 7 " ... " } {XPPEDIT 18 0 "matrix([[1, 2, -1], [0, -8, 3], [0, 0,0]])" "6#-%'matri xG6#7%7%\"\"\"\"\"#,$F(!\"\"7%\"\"!,$\"\")F+\"\"$7%F-F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[2] = R[2]*``(-1/8);" "6#/&%\"RG6#\"\"#*&&F%6#F'\" \"\"-%!G6#,$*&F+F+\"\")!\"\"F2F+" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[1, 2, -1], [0, 1, -3/8], [0, 0,0]])" "6#-%'matrixG6#7%7%\" \"\"\"\"#,$F(!\"\"7%\"\"!F(,$*&\"\"$F(\"\")F+F+7%F-F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[1] = R[1]-2*R[2];" "6#/&%\"RG6#\"\"\",&&F%6#F'F'*&\" \"#F'&F%6#F,F'!\"\"" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[1, 0, -1/4], [0, 1, -3/8], [0, 0,0]])" "6#-%'matrixG6#7%7%\"\"\"\"\"!,$* &F(F(\"\"%!\"\"F-7%F)F(,$*&\"\"$F(\"\")F-F-7%F)F)F)" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "PIECEWISE([x-z/4 = 0, ``],[y-3/8*z = 0, ``]);" "6#-%*PI ECEWISEG6$7$/,&%\"xG\"\"\"*&%\"zGF*\"\"%!\"\"F.\"\"!%!G7$/,&%\"yGF**( \"\"$F*\"\")F.F,F*F.F/F0" }{TEXT -1 5 "or " }{XPPEDIT 18 0 "PIECEWIS E([x=z/4, ``],[y=3/8*z, ``])" "6#-%*PIECEWISEG6$7$/%\"xG*&%\"zG\"\"\" \"\"%!\"\"%!G7$/%\"yG*(\"\"$F+\"\")F-F*F+F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "z = 8*t;" "6#/%\"zG*&\"\") \"\"\"%\"tGF'" }{TEXT -1 7 ". Then " }{XPPEDIT 18 0 "x = 2*t;" "6#/%\" xG*&\"\"#\"\"\"%\"tGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 3*t;" "6#/%\"yG*&\"\"$\"\"\"%\"tGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "There are infinitely many solutions given by the parametr ic equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIE CEWISE([x = 2*t, ``],[y = 3*t, ``],[z = 8*t, ``]);" "6#-%*PIECEWISEG6% 7$/%\"xG*&\"\"#\"\"\"%\"tGF+%!G7$/%\"yG*&\"\"$F+F,F+F-7$/%\"zG*&\"\")F +F,F+F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 72 "These equatio ns can be written in the form of a single vector equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x], [y], [z]]) = t *matrix([[2], [3], [8]]);" "6#/-%'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG*&% \"tG\"\"\"-F%6#7%7#\"\"#7#\"\"$7#\"\")F0" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 44 "Such vectors are eigenvectors provided that " } {XPPEDIT 18 0 "t <>0" "6#0%\"tG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "As a check note that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[3, 2, -2], [-1, -4, 1], [2, -4, -1]])*m atrix([[2], [3], [8]])=matrix([[-4], [-6], [-16]])" "6#/*&-%'matrixG6# 7%7%\"\"$\"\"#,$F+!\"\"7%,$\"\"\"F-,$\"\"%F-F07%F+,$F2F-,$F0F-F0-F&6#7 %7#F+7#F*7#\"\")F0-F&6#7%7#,$F2F-7#,$\"\"'F-7#,$\"#;F-" }{XPPEDIT 18 0 "``=-2*matrix([[2], [3], [8]])" "6#/%!G,$*&\"\"#\"\"\"-%'matrixG6#7% 7#F'7#\"\"$7#\"\")F(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 44 "Eigenvectors associated with t he eigenvalue " }{XPPEDIT 18 0 "lambda = 1;" "6#/%'lambdaG\"\"\"" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "Consider the equation \+ " }{XPPEDIT 18 0 "(A-I)*`.`*X = ``;" "6#/*(,&%\"AG\"\"\"%\"IG!\"\"F'% \".GF'%\"XGF'%!G" }{TEXT 294 1 "0" }{TEXT -1 11 ", that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[2, 2, -2], [-1, -5 , 1], [2, -4, -2]])*matrix([[x], [y], [z]]) = matrix([[0], [0], [0]]); " "6#/*&-%'matrixG6#7%7%\"\"#F*,$F*!\"\"7%,$\"\"\"F,,$\"\"&F,F/7%F*,$ \"\"%F,,$F*F,F/-F&6#7%7#%\"xG7#%\"yG7#%\"zGF/-F&6#7%7#\"\"!7#FC7#FC" } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "X=matrix([[x],[y],[z]])" "6#/%\" XG-%'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We solve this system of equations by applying row operations to the coefficient matrix:" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[2, 2, -2], \+ [-1, -5, 1], [2, -4, -2]])" "6#-%'matrixG6#7%7%\"\"#F(,$F(!\"\"7%,$\" \"\"F*,$\"\"&F*F-7%F(,$\"\"%F*,$F(F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIE CEWISE([R[1]=R[1]*`.`*``(1/2),``],[R[3]=R[3]*`.`*``(1/2),``])" "6#-%*P IECEWISEG6$7$/&%\"RG6#\"\"\"*(&F)6#F+F+%\".GF+-%!G6#*&F+F+\"\"#!\"\"F+ F17$/&F)6#\"\"$*(&F)6#F:F+F/F+-F16#*&F+F+F4F5F+F1" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[1, 1, -1], [-1, -5, 1], [1, -2, -1]]);" "6 #-%'matrixG6#7%7%\"\"\"F(,$F(!\"\"7%,$F(F*,$\"\"&F*F(7%F(,$\"\"#F*,$F( F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R[2] = R[2]+R[1], ``],[R[3 ] = R[3]-R[1], ``]);" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"#,&&F)6#F+\"\" \"&F)6#F/F/%!G7$/&F)6#\"\"$,&&F)6#F7F/&F)6#F/!\"\"F2" }{TEXT -1 7 " \+ ... " }{XPPEDIT 18 0 "matrix([[1, 1, -1], [0, -4, 0], [0, -3, 0]]);" " 6#-%'matrixG6#7%7%\"\"\"F(,$F(!\"\"7%\"\"!,$\"\"%F*F,7%F,,$\"\"$F*F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R[2] = R[2]*`.`*``(-1/4), `` ],[R[3] = R[3]*`.`*``(-1/3), ``]);" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"# *(&F)6#F+\"\"\"%\".GF/-%!G6#,$*&F/F/\"\"%!\"\"F7F/F27$/&F)6#\"\"$*(&F) 6#F0" "6#0%\"tG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "As a check note that: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[3, 2, -2], [ -1, -4, 1], [2, -4, -1]])*matrix([[1], [0], [1]]) = matrix([[1], [0], \+ [1]]);" "6#/*&-%'matrixG6#7%7%\"\"$\"\"#,$F+!\"\"7%,$\"\"\"F-,$\"\"%F- F07%F+,$F2F-,$F0F-F0-F&6#7%7#F07#\"\"!7#F0F0-F&6#7%7#F07#F;7#F0" } {TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 44 "Eigenvectors associated with the eigenvalue " }{XPPEDIT 18 0 "lambda = -1;" "6#/%'lambdaG,$\"\"\"!\"\"" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 23 "Consider the equation " }{XPPEDIT 18 0 " (A+I)*`.`*X = ``;" "6#/*(,&%\"AG\"\"\"%\"IGF'F'%\".GF'%\"XGF'%!G" } {TEXT 295 1 "0" }{TEXT -1 11 ", that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[4, 2, -2], [-1, -3, 1], [2, -4, 0]]) *matrix([[x], [y], [z]]) = matrix([[0], [0], [0]]);" "6#/*&-%'matrixG6 #7%7%\"\"%\"\"#,$F+!\"\"7%,$\"\"\"F-,$\"\"$F-F07%F+,$F*F-\"\"!F0-F&6#7 %7#%\"xG7#%\"yG7#%\"zGF0-F&6#7%7#F57#F57#F5" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "X=matrix([[x],[y],[z]])" "6#/%\"XG-%'matrixG6#7%7#%\"xG 7#%\"yG7#%\"zG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 87 "We solve this system of equations by appl ying row operations to the coefficient matrix:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[4, 2, -2], [-1, -3, 1], [2, -4 , 0]]);" "6#-%'matrixG6#7%7%\"\"%\"\"#,$F)!\"\"7%,$\"\"\"F+,$\"\"$F+F. 7%F),$F(F+\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R[1]=R[1]* `.`*``(1/2),``],[R[3]=R[3]*`.`*``(1/2),``])" "6#-%*PIECEWISEG6$7$/&%\" RG6#\"\"\"*(&F)6#F+F+%\".GF+-%!G6#*&F+F+\"\"#!\"\"F+F17$/&F)6#\"\"$*(& F)6#F:F+F/F+-F16#*&F+F+F4F5F+F1" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[2, 1, -1], [-1, -3, 1], [1, -2, 0]]);" "6#-%'matrixG6#7%7%\" \"#\"\"\",$F)!\"\"7%,$F)F+,$\"\"$F+F)7%F),$F(F+\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 "swap \+ " }{XPPEDIT 18 0 "R[1];" "6#&%\"RG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "R[3]" "6#&%\"RG6#\"\"$" }{TEXT -1 7 " ... " } {XPPEDIT 18 0 "matrix([[1, -2, 0], [-1, -3, 1], [2, 1, -1]]);" "6#-%'m atrixG6#7%7%\"\"\",$\"\"#!\"\"\"\"!7%,$F(F+,$\"\"$F+F(7%F*F(,$F(F+" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R[2] = R[2]+R[1], ``],[R[3] \+ = R[3]-2*R[1], ``]);" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"#,&&F)6#F+\"\" \"&F)6#F/F/%!G7$/&F)6#\"\"$,&&F)6#F7F/*&F+F/&F)6#F/F/!\"\"F2" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[1, -2, 0], [0, -5, 1], [0, 5, \+ -1]]);" "6#-%'matrixG6#7%7%\"\"\",$\"\"#!\"\"\"\"!7%F,,$\"\"&F+F(7%F,F /,$F(F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[3] = R[3]+R[2];" "6#/&%\"RG6# \"\"$,&&F%6#F'\"\"\"&F%6#\"\"#F+" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[1, -2, 0], [0, -5, 1], [0, 0, 0]]);" "6#-%'matrixG6#7%7%\" \"\",$\"\"#!\"\"\"\"!7%F,,$\"\"&F+F(7%F,F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[2] = R[2]*`.`*``(-1/5);" "6#/&%\"RG6#\"\"#*(&F%6#F'\"\"\"%\".G F+-%!G6#,$*&F+F+\"\"&!\"\"F3F+" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 " matrix([[1, -2, 0], [0, 1, -1/5], [0, 0, 0]]);" "6#-%'matrixG6#7%7%\" \"\",$\"\"#!\"\"\"\"!7%F,F(,$*&F(F(\"\"&F+F+7%F,F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R[1] = R[1]+2*R[2];" "6#/&%\"RG6#\"\"\",&&F%6#F'F'*&\" \"#F'&F%6#F,F'F'" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[1, 0, -2/5], [0, 1, -1/5], [0, 0, 0]]);" "6#-%'matrixG6#7%7%\"\"\"\"\"!,$*& \"\"#F(\"\"&!\"\"F.7%F)F(,$*&F(F(F-F.F.7%F)F)F)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "PIECEWISE([x-2/5*z = 0, ``],[y-z/5 = 0, ``]);" "6#-% *PIECEWISEG6$7$/,&%\"xG\"\"\"*(\"\"#F*\"\"&!\"\"%\"zGF*F.\"\"!%!G7$/,& %\"yGF**&F/F*F-F.F.F0F1" }{TEXT -1 5 "or " }{XPPEDIT 18 0 "PIECEWISE ([x=2/5*z, ``],[y=z/5, ``])" "6#-%*PIECEWISEG6$7$/%\"xG*(\"\"#\"\"\"\" \"&!\"\"%\"zGF+%!G7$/%\"yG*&F.F+F,F-F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "z = 5*t;" "6#/%\"zG*&\"\"&\"\" \"%\"tGF'" }{TEXT -1 7 ". Then " }{XPPEDIT 18 0 "x = 2*t;" "6#/%\"xG*& \"\"#\"\"\"%\"tGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = t;" "6#/% \"yG%\"tG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "There are i nfinitely many solutions given by the parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = 2*t, ``],[y \+ = t, ``],[z = 5*t, ``]);" "6#-%*PIECEWISEG6%7$/%\"xG*&\"\"#\"\"\"%\"tG F+%!G7$/%\"yGF,F-7$/%\"zG*&\"\"&F+F,F+F-" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 72 "These equations can be written in the form of a si ngle vector equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x], [y], [z]]) = t*matrix([[2], [1], [5]]);" "6#/-%'mat rixG6#7%7#%\"xG7#%\"yG7#%\"zG*&%\"tG\"\"\"-F%6#7%7#\"\"#7#F07#\"\"&F0 " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Such vectors are eig envectors provided that " }{XPPEDIT 18 0 "t <>0" "6#0%\"tG\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "As a check note that: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[3, 2, -2 ], [-1, -4, 1], [2, -4, -1]])*matrix([[2], [1], [5]]) = matrix([[-2], \+ [-1], [-5]]);" "6#/*&-%'matrixG6#7%7%\"\"$\"\"#,$F+!\"\"7%,$\"\"\"F-,$ \"\"%F-F07%F+,$F2F-,$F0F-F0-F&6#7%7#F+7#F07#\"\"&F0-F&6#7%7#,$F+F-7#,$ F0F-7#,$F0" "6%/ %'lambdaG,$\"\"#!\"\"*&%\"tG\"\"\"-%'matrixG6#7%7#F&7#\"\"$7#\"\")F*0F )\"\"!" }{TEXT -1 5 "; " }{XPPEDIT 18 0 "lambda = 1,t*matrix([[1], \+ [0], [1]]),t <> 0;" "6%/%'lambdaG\"\"\"*&%\"tGF%-%'matrixG6#7%7#F%7#\" \"!7#F%F%0F'F." }{TEXT -1 6 "; " }{XPPEDIT 18 0 "lambda = -1,t*mat rix([[2], [1], [5]]),t <> 0;" "6%/%'lambdaG,$\"\"\"!\"\"*&%\"tGF&-%'ma trixG6#7%7#\"\"#7#F&7#\"\"&F&0F)\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "A := ma trix([[3,2,-2],[-1,-4,1],[2,-4,-1]]);\nlinalg[eigenvalues](A);\nlinalg [eigenvectors](A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG 6#7%7%\"\"$\"\"#!\"#7%!\"\"!\"%\"\"\"7%F+F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6%!\"\"\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%! \"#\"\"\"<#-%'vectorG6#7%#\"\"#\"\"$F%#\"\")F-7%!\"\"F%<#-F(6#7%F,F%\" \"&7%F%F%<#-F(6#7%F%\"\"!F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "A := Matrix([[3,2,-2],[-1,-4 ,1],[2,-4,-1]]);\nLinearAlgebra[Eigenvectors](A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*K9>X\"-%'MATRIXG6#7%7%\"\"$\"\"#! \"#7%!\"\"!\"%\"\"\"7%F/F3F2%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"*OY,X\"-%'MATRIXG6#7%7#!\"#7#\"\"\"7#!\"\"&%'VectorG6 #%'columnG-F$6%\"*_%)>X\"-F(6#7%7%F.F.\"\"#7%#\"\"$F<\"\"!F.7%\"\"%F. \"\"&%'MatrixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Exa mple 2 " }}{PARA 0 "" 0 "" {TEXT 296 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 52 "Find the eigenvalues and eigenvectors of \+ the matrix " }{XPPEDIT 18 0 "A=matrix([[21,-60],[8,-23]])" "6#/%\"AG-% 'matrixG6#7$7$\"#@,$\"#g!\"\"7$\"\"),$\"#BF-" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT 297 8 "Solution" }{TEXT -1 2 ": " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A-lambda*I=matrix([[21-lambda, - 60], [8, -23-lambda]])" "6#/,&%\"AG\"\"\"*&%'lambdaGF&%\"IGF&!\"\"-%'m atrixG6#7$7$,&\"#@F&F(F*,$\"#gF*7$\"\"),&\"#BF*F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "The characteristic equation of A is: " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "det(A-lambda*I)=0" "6#/-%$detG6#,&%\"AG\"\"\"*&%'lambda GF)%\"IGF)!\"\"\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "t hat is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 173 99 99 {PLOTDATA 2 "6,-%'CURVESG6$7$7$$\"3++++++++D!#<$!35+++++++?!#=7$F($\"3 %**************>\"F*-%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$\"3++++++++XF *F+7$F:F/F1-%%TEXTG6&7$$\"\"$F5$\"\"\"F5Q'21~-~l6\"F1-%%FONTG6$%'SYMBO LG\"#7-F>6&7$$\"\"%F5FCQ$-60FFF1FG-F>6&7$FA$F5F5Q\"8FFF1FG-F>6&7$FOFUQ (-23~-~lFFF1FG-F>6&7$$\"$&[!\"#$\"\"&!\"\"Q%=~0.FFF1FG-%+AXESLABELSG6$ Q!FFFbo-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAULTGFjo" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 58 "The characteristic equation can be simplified as f ollows. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(21-lambd a)*(-23-lambda)+480=0" "6#/,&*&,&\"#@\"\"\"%'lambdaG!\"\"F(,&\"#BF*F)F *F(F(\"$![F(\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda^2+23*lambda-21*lambda-483+480=0" "6#/,,*$%'lamb daG\"\"#\"\"\"*&\"#BF(F&F(F(*&\"#@F(F&F(!\"\"\"$$[F-\"$![F(\"\"!" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lam bda^2+2*lambda-3=0" "6#/,(*$%'lambdaG\"\"#\"\"\"*&F'F(F&F(F(\"\"$!\"\" \"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(lambda+3)*(lambda-1)=0" "6#/*&,&%'lambdaG\"\"\"\"\"$F'F',&F&F'F '!\"\"F'\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The eig envalues are therefore " }{XPPEDIT 18 0 "lambda=-3" "6#/%'lambdaG,$\" \"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "lambda=1" "6#/%'lambdaG \"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 " " 0 "" {TEXT -1 44 "Eigenvectors associated with the eigenvalue " } {XPPEDIT 18 0 "lambda = -3;" "6#/%'lambdaG,$\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "Consider the equation " }{XPPEDIT 18 0 "(A+3*I)*`.`*X = ``;" "6#/*(,&%\"AG\"\"\"*&\"\"$F'%\"IGF'F'F'%\". GF'%\"XGF'%!G" }{TEXT 298 1 "0" }{TEXT -1 11 ", that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[24, -60], [8, -20] ])*matrix([[x], [y]]) = matrix([[0], [0]]);" "6#/*&-%'matrixG6#7$7$\"# C,$\"#g!\"\"7$\"\"),$\"#?F-\"\"\"-F&6#7$7#%\"xG7#%\"yGF2-F&6#7$7#\"\"! 7#F>" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "X=matrix([[x], [y]])" "6# /%\"XG-%'matrixG6#7$7#%\"xG7#%\"yG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 31 "We obtain the single equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "8*x-20*y=0" "6#/,&*&\"\")\"\"\"%\"xGF'F '*&\"#?F'%\"yGF'!\"\"\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y \+ = 2/5*x;" "6#/%\"yG*(\"\"#\"\"\"\"\"&!\"\"%\"xGF'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "x=5*t" "6#/%\"xG*&\" \"&\"\"\"%\"tGF'" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "y=2*t" "6# /%\"yG*&\"\"#\"\"\"%\"tGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "The eigenvectors associated with the eigenvalue " }{XPPEDIT 18 0 "lambda = -3;" "6#/%'lambdaG,$\"\"$!\"\"" }{TEXT -1 5 " are " } {XPPEDIT 18 0 "t*matrix([[5],[2]])" "6#*&%\"tG\"\"\"-%'matrixG6#7$7#\" \"&7#\"\"#F%" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "t<>0" "6#0%\"tG\" \"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 44 "Eigenvectors associated with the eigenvalue " } {XPPEDIT 18 0 "lambda = 1;" "6#/%'lambdaG\"\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 23 "Consider the equation " }{XPPEDIT 18 0 " (A-I)*`.`*X = ``;" "6#/*(,&%\"AG\"\"\"%\"IG!\"\"F'%\".GF'%\"XGF'%!G" } {TEXT 299 1 "0" }{TEXT -1 11 ", that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[20, -60], [8, -24]])*matrix([[x], [y ]]) = matrix([[0], [0]]);" "6#/*&-%'matrixG6#7$7$\"#?,$\"#g!\"\"7$\"\" ),$\"#CF-\"\"\"-F&6#7$7#%\"xG7#%\"yGF2-F&6#7$7#\"\"!7#F>" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "X=matrix([[x], [y]])" "6#/%\"XG-%'matrixG6# 7$7#%\"xG7#%\"yG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "We \+ obtain the single equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x = 3*y;" "6#/%\"xG*&\"\"$\"\"\"%\"yGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "y = t;" "6#/%\" yG%\"tG" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "y = 3*t;" "6#/%\"yG *&\"\"$\"\"\"%\"tGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 " The eigenvectors associated with the eigenvalue " }{XPPEDIT 18 0 "lamb da = 1;" "6#/%'lambdaG\"\"\"" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "t*ma trix([[3], [1]]);" "6#*&%\"tG\"\"\"-%'matrixG6#7$7#\"\"$7#F%F%" } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "t<>0" "6#0%\"tG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "The following two matrix multipl ications check the results: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[21, -60], [8, -23]])*matrix([[5], [2]])=matrix ([[-15], [-6]])" "6#/*&-%'matrixG6#7$7$\"#@,$\"#g!\"\"7$\"\"),$\"#BF- \"\"\"-F&6#7$7#\"\"&7#\"\"#F2-F&6#7$7#,$\"#:F-7#,$\"\"'F-" }{XPPEDIT 18 0 "``=-3*matrix([[5], [2]])" "6#/%!G,$*&\"\"$\"\"\"-%'matrixG6#7$7# \"\"&7#\"\"#F(!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "matrix([[21, -60], [8, -23]])*matrix([[3], [1]])=matrix([[3], [1]])" "6#/*&-%'matr ixG6#7$7$\"#@,$\"#g!\"\"7$\"\"),$\"#BF-\"\"\"-F&6#7$7#\"\"$7#F2F2-F&6# 7$7#F77#F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "A := Matrix([[21,-60],[8,-23]]);\nL inearAlgebra[Eigenvectors](A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" AG-%'RTABLEG6%\"*w**>X\"-%'MATRIXG6#7$7$\"#@!#g7$\"\")!#B%'MatrixG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"*'z9]9-%'MATRIXG6#7$7#! \"$7#\"\"\"&%'VectorG6#%'columnG-F$6%\"*sj?X\"-F(6#7$7$#\"\"&\"\"#\"\" $7$F.F.%'MatrixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Ex ample 3 " }}{PARA 0 "" 0 "" {TEXT 300 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 52 "Find the eigenvalues and eigenvectors of \+ the matrix " }{XPPEDIT 18 0 "A=matrix([[1,2,1], [6,-1,0], [-1,-2,-1]]) " "6#/%\"AG-%'matrixG6#7%7%\"\"\"\"\"#F*7%\"\"',$F*!\"\"\"\"!7%,$F*F/, $F+F/,$F*F/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT 301 8 "Solution " }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " A-lambda*I = matrix([[1, 2, 1], [6, -1, 0], [-1, -2, -1]])-lambda*matr ix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])" "6#/,&%\"AG\"\"\"*&%'lambdaGF&% \"IGF&!\"\",&-%'matrixG6#7%7%F&\"\"#F&7%\"\"',$F&F*\"\"!7%,$F&F*,$F1F* ,$F&F*F&*&F(F&-F-6#7%7%F&F5F57%F5F&F57%F5F5F&F&F*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = matrix([[1-lambda, 2, 1], [6, -1-lambda, 0], [-1, \+ -2, -1-lambda]])" "6#/%!G-%'matrixG6#7%7%,&\"\"\"F+%'lambdaG!\"\"\"\"# F+7%\"\"',&F+F-F,F-\"\"!7%,$F+F-,$F.F-,&F+F-F,F-" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The chara cteristic equation of " }{TEXT 302 1 "A" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 221 125 125 {PLOTDATA 2 "61-%'CU RVESG6$7$7$$!3A+++++++S!#=$!35+++++++?F*7$F($\"33+++++++@!#<-%'COLOURG 6&%$RGBG\"\"!F5F5-F$6$7$7$$\"3!**************R#F0F+7$F:F.F1-%%TEXTG6&7 $$F5F5$\"\"#F5Q$1-l6\"F1-%%FONTG6$%'SYMBOLG\"#7-F>6&7$$\"\"\"F5FBQ\"2F EF1FF-F>6&7$FBFBQ\"1FEF1FF-F>6&7$FAFNQ\"6FEF1FF-F>6&7$FNFNQ%-1-lFEF1FF -F>6&7$FBFNQ\"0FEF1FF-F>6&7$FAFAQ#-1FEF1FF-F>6&7$FNFAQ#-2FEF1FF-F>6&7$ FBFAFfnF1FF-F>6%7$$\"#G!\"\"FNQ%=~0.FE-FG6%%&TIMESG%&ROMANGFJ-%+AXESLA BELSG6$Q!FEFdp-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAULTGF\\q" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "C urve 10" "Curve 11" "Curve 12" }}{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 54 "Expanding the determinant along the second row gives: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 370 91 91 {PLOTDATA 2 "64 -%'CURVESG6$7$7$$\"3k*************4%!#<$\"3))**************H!#=7$F($\" 33+++++++;F*-%'COLOURG6&%$RGBG\"\"!F5F5-F$6$7$7$$\"3M+++++++fF*F+7$F:F /F1-F$6$7$7$$\"3M+++++++uF*F+7$FAF/F1-F$6$7$7$$\"3G*************>*F*F+ 7$FHF/F1-%%TEXTG6&7$$\"#X!\"\"$\"#:FQQ\"26\"F1-%%FONTG6$%'SYMBOLG\"#7- FL6&7$$\"#bFQFRQ\"1FUF1FV-FL6&7$FO$\"\"&FQQ#-2FUF1FV-FL6&7$$\"#aFQF^oQ '-1~-~lFUF1FV-FL6&7$$\"#yFQFRQ&1~-~lFUF1FV-FL6&7$$\"#))FQFRFjnF1FV-FL6 &7$FjoF^oQ#-1FUF1FV-FL6&7$$\"#()FQF^oFfoF1FV-FL6&7$$\"#PFQ$\"\"\"F5Q'( ~-6~)FUF1FV-FL6&7$$\"#nFQF`qQ-+~(~-1~-~l~)FUF1FV-FL6&7$$\"#&*FQF`qQ%=~ 0.FUF1FV-%+AXESLABELSG6$Q!FUFbr-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAU LTGFjr" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" " Curve 15" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 58 "The characte ristic equation can be simplified as follows. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(-6)*(-2-2*lambda+2)+(-1-lambda)*((1-la mbda)*(-1-lambda)+1)=0" "6#/,&*&,$\"\"'!\"\"\"\"\",(\"\"#F(*&F+F)%'lam bdaGF)F(F+F)F)F)*&,&F)F(F-F(F),&*&,&F)F)F-F(F),&F)F(F-F(F)F)F)F)F)F)\" \"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "12*lambda-(lambda+1)*((lambda-1)*(la mbda+1)+1)=0" "6#/,&*&\"#7\"\"\"%'lambdaGF'F'*&,&F(F'F'F'F',&*&,&F(F'F '!\"\"F',&F(F'F'F'F'F'F'F'F'F.\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "12*lambda-(lambda+1)*(lambda^2-1+1)= 0" "6#/,&*&\"#7\"\"\"%'lambdaGF'F'*&,&F(F'F'F'F',(*$F(\"\"#F'F'!\"\"F' F'F'F.\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "12*lambda-(lambda+1)*lambda^2=0" "6#/,&*&\"#7\"\"\"%'la mbdaGF'F'*&,&F(F'F'F'F'*$F(\"\"#F'!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "12*lambda-lambda^3-lambda ^2=0" "6#/,(*&\"#7\"\"\"%'lambdaGF'F'*$F(\"\"$!\"\"*$F(\"\"#F+\"\"!" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lamb da^3+lambda^2-12*lambda=0" "6#/,(*$%'lambdaG\"\"$\"\"\"*$F&\"\"#F(*&\" #7F(F&F(!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "lambda*(lambda^2+lambda-12)=0" "6#/*&%'lambdaG\"\"\" ,(*$F%\"\"#F&F%F&\"#7!\"\"F&\"\"!" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda*(lambda+4)*(lambda-3)=0" "6#/* (%'lambdaG\"\"\",&F%F&\"\"%F&F&,&F%F&\"\"$!\"\"F&\"\"!" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The e igenvalues are " }{XPPEDIT 18 0 "lambda = 0;" "6#/%'lambdaG\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "lambda = -4;" "6#/%'lambdaG,$\"\"%!\" \"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "lambda = 3;" "6#/%'lambdaG\"\" $" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 " " {TEXT -1 44 "Eigenvectors associated with the eigenvalue " } {XPPEDIT 18 0 "lambda = 0;" "6#/%'lambdaG\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 23 "Consider the equation " }{XPPEDIT 18 0 " A*`.`*X = ``;" "6#/*(%\"AG\"\"\"%\".GF&%\"XGF&%!G" }{TEXT 303 1 "0" } {TEXT -1 11 ", that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[1, 2, 1], [6, -1, 0], [-1, -2, -1]])*matrix([[ x], [y], [z]]) = matrix([[0], [0], [0]])" "6#/*&-%'matrixG6#7%7%\"\"\" \"\"#F*7%\"\"',$F*!\"\"\"\"!7%,$F*F/,$F+F/,$F*F/F*-F&6#7%7#%\"xG7#%\"y G7#%\"zGF*-F&6#7%7#F07#F07#F0" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 " X=matrix([[x],[y],[z]])" "6#/%\"XG-%'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 88 "We solve this system of equations by applying row operati ons to the coefficient matrix: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[1, 2, 1], [6, -1, 0], [-1, -2, -1]])" "6#-%'ma trixG6#7%7%\"\"\"\"\"#F(7%\"\"',$F(!\"\"\"\"!7%,$F(F-,$F)F-,$F(F-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R[2] = R[2]-6*R[1], ``],[R[3 ] = R[3]+R[1], ``]);" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"#,&&F)6#F+\"\" \"*&\"\"'F/&F)6#F/F/!\"\"%!G7$/&F)6#\"\"$,&&F)6#F:F/&F)6#F/F/F5" } {TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[1, 2, 1], [0, -13, -6], \+ [0,0,0]])" "6#-%'matrixG6#7%7%\"\"\"\"\"#F(7%\"\"!,$\"#8!\"\",$\"\"'F. 7%F+F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[2] = R[2]*`.`*``(-1/13);" "6#/&%\"RG6#\"\"#*(&F%6#F'\"\"\"%\".GF+ -%!G6#,$*&F+F+\"#8!\"\"F3F+" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "mat rix([[1, 2, 1], [0, 1, 6/13], [0, 0, 0]])" "6#-%'matrixG6#7%7%\"\"\"\" \"#F(7%\"\"!F(*&\"\"'F(\"#8!\"\"7%F+F+F+" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[1] = R[1]-2*R[2];" "6#/&%\"RG6#\"\"\",&&F%6#F'F'*&\"\"#F'&F%6#F,F '!\"\"" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[1, 0, 1/13], [0 , 1, 6/13], [0, 0, 0]])" "6#-%'matrixG6#7%7%\"\"\"\"\"!*&F(F(\"#8!\"\" 7%F)F(*&\"\"'F(F+F,7%F)F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "PIECEWISE([x+z/ 13 = 0, ``],[y+6/13*z = 0, ``]);" "6#-%*PIECEWISEG6$7$/,&%\"xG\"\"\"*& %\"zGF*\"#8!\"\"F*\"\"!%!G7$/,&%\"yGF**(\"\"'F*F-F.F,F*F*F/F0" }{TEXT -1 7 " or " }{XPPEDIT 18 0 "PIECEWISE([z=-13*x, ``],[13*y=-6*z, ``] )" "6#-%*PIECEWISEG6$7$/%\"zG,$*&\"#8\"\"\"%\"xGF,!\"\"%!G7$/*&F+F,%\" yGF,,$*&\"\"'F,F(F,F.F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "x = t;" "6#/%\"xG%\"tG" }{TEXT -1 7 ". Then \+ " }{XPPEDIT 18 0 "z = -13*t;" "6#/%\"zG,$*&\"#8\"\"\"%\"tGF(!\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 6*t;" "6#/%\"yG*&\"\"'\"\"\"%\" tGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "There are infini tely many solutions given by the parametric equations: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = t, ``],[y = 6*t, \+ ``],[z = -13*t, ``]);" "6#-%*PIECEWISEG6%7$/%\"xG%\"tG%!G7$/%\"yG*&\" \"'\"\"\"F)F0F*7$/%\"zG,$*&\"#8F0F)F0!\"\"F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 72 "These equations can be written in the for m of a single vector equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[x], [y], [z]]) = t*matrix([[1], [6], [-13]]); " "6#/-%'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG*&%\"tG\"\"\"-F%6#7%7#F07#\" \"'7#,$\"#8!\"\"F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Su ch vectors are eigenvectors provided that " }{XPPEDIT 18 0 "t <>0" "6# 0%\"tG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "As a chec k note that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matr ix([[1, 2, 1], [6, -1, 0], [-1, -2, -1]])*matrix([[1], [6], [-13]]) = \+ matrix([[0], [0], [0]])" "6#/*&-%'matrixG6#7%7%\"\"\"\"\"#F*7%\"\"',$F *!\"\"\"\"!7%,$F*F/,$F+F/,$F*F/F*-F&6#7%7#F*7#F-7#,$\"#8F/F*-F&6#7%7#F 07#F07#F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 44 "Eigenvectors associated with the eigenvalue " }{XPPEDIT 18 0 "lambda = -4;" "6#/%'lambdaG,$\"\"%!\"\"" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 23 "Consider the equation " } {XPPEDIT 18 0 "(A+4*I)*`.`*X = ``;" "6#/*(,&%\"AG\"\"\"*&\"\"%F'%\"IGF 'F'F'%\".GF'%\"XGF'%!G" }{TEXT 304 1 "0" }{TEXT -1 11 ", that is, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[5, 2, 1], [6 , 3, 0], [-1, -2, 3]])*matrix([[x], [y], [z]]) = matrix([[0], [0], [0] ]);" "6#/*&-%'matrixG6#7%7%\"\"&\"\"#\"\"\"7%\"\"'\"\"$\"\"!7%,$F,!\" \",$F+F3F/F,-F&6#7%7#%\"xG7#%\"yG7#%\"zGF,-F&6#7%7#F07#F07#F0" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "X=matrix([[x],[y],[z]])" "6#/%\"XG-%'m atrixG6#7%7#%\"xG7#%\"yG7#%\"zG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 88 "We solve this system of \+ equations by applying row operations to the coefficient matrix: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[5, 2, 1], [6 , 3, 0], [-1, -2, 3]])" "6#-%'matrixG6#7%7%\"\"&\"\"#\"\"\"7%\"\"'\"\" $\"\"!7%,$F*!\"\",$F)F1F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 6 " swap " }{XPPEDIT 18 0 "R[1]" "6#&%\"RG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "R[3]" "6#&%\"RG 6#\"\"$" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[-1, -2, 3], [6 , 3, 0],[5, 2, 1]])" "6#-%'matrixG6#7%7%,$\"\"\"!\"\",$\"\"#F*\"\"$7% \"\"'F-\"\"!7%\"\"&F,F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R[1] = -R[1]" "6 #/&%\"RG6#\"\"\",$&F%6#F'!\"\"" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 " matrix([[1, 2, -3], [6, 3, 0], [5, 2, 1]])" "6#-%'matrixG6#7%7%\"\"\" \"\"#,$\"\"$!\"\"7%\"\"'F+\"\"!7%\"\"&F)F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R[2] = R[2]-6*R[1], ``],[R[3] = R[3]-5*R[1], ``]);" "6#- %*PIECEWISEG6$7$/&%\"RG6#\"\"#,&&F)6#F+\"\"\"*&\"\"'F/&F)6#F/F/!\"\"%! G7$/&F)6#\"\"$,&&F)6#F:F/*&\"\"&F/&F)6#F/F/F4F5" }{TEXT -1 7 " ... \+ " }{XPPEDIT 18 0 "matrix([[1, 2, -3],[0, -9, 18], [0, -8, 16]])" "6#-% 'matrixG6#7%7%\"\"\"\"\"#,$\"\"$!\"\"7%\"\"!,$\"\"*F,\"#=7%F.,$\"\")F, \"#;" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R[2] = R[2]*`.`*``(-1/9 ), ``],[R[2] = R[2]*`.`*``(-1/8), ``]);" "6#-%*PIECEWISEG6$7$/&%\"RG6# \"\"#*(&F)6#F+\"\"\"%\".GF/-%!G6#,$*&F/F/\"\"*!\"\"F7F/F27$/&F)6#F+*(& F)6#F+F/F0F/-F26#,$*&F/F/\"\")F7F7F/F2" }{TEXT -1 7 " ... " } {XPPEDIT 18 0 "matrix([[1, 2, -3], [0, 1,-2], [0, 1,-2]])" "6#-%'matri xG6#7%7%\"\"\"\"\"#,$\"\"$!\"\"7%\"\"!F(,$F)F,7%F.F(,$F)F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE(([R[1] = R[1]-2*R[2],``],[R[3] = R[3]-R[2 ],``])" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"\",&&F)6#F+F+*&\"\"#F+&F)6#F0 F+!\"\"%!G7$/&F)6#\"\"$,&&F)6#F9F+&F)6#F0F3F4" }{TEXT -1 7 " ... " } {XPPEDIT 18 0 "matrix([[1, 0, 1], [0, 1,-2], [0, 0,0]])" "6#-%'matrixG 6#7%7%\"\"\"\"\"!F(7%F)F(,$\"\"#!\"\"7%F)F)F)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "This give s " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "PIECEWISE([x+z = 0, ``],[y-2*z \+ = 0, ``]);" "6#-%*PIECEWISEG6$7$/,&%\"xG\"\"\"%\"zGF*\"\"!%!G7$/,&%\"y GF**&\"\"#F*F+F*!\"\"F,F-" }{TEXT -1 7 " or " }{XPPEDIT 18 0 "PIECE WISE([x = -z, ``],[y = 2*z, ``]);" "6#-%*PIECEWISEG6$7$/%\"xG,$%\"zG! \"\"%!G7$/%\"yG*&\"\"#\"\"\"F*F2F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 4 "Let " }{XPPEDIT 18 0 "z = t;" "6#/%\"zG%\"tG" }{TEXT -1 7 ". Then " }{XPPEDIT 18 0 "x = -t;" "6#/%\"xG,$%\"tG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 2*t;" "6#/%\"yG*&\"\"#\"\"\"%\"tGF'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "There are infinitely ma ny solutions given by the parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = -t, ``],[y = 2*t, ``],[z = t, ``]);" "6#-%*PIECEWISEG6%7$/%\"xG,$%\"tG!\"\"%!G7$/%\"yG*&\"\"# \"\"\"F*F2F,7$/%\"zGF*F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 72 "These equations can be written in the form of a single vector equa tion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x] , [y], [z]]) = t*matrix([[-1], [2], [1]]);" "6#/-%'matrixG6#7%7#%\"xG7 #%\"yG7#%\"zG*&%\"tG\"\"\"-F%6#7%7#,$F0!\"\"7#\"\"#7#F0F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Such vectors are eigenvectors pr ovided that " }{XPPEDIT 18 0 "t <>0" "6#0%\"tG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "As a check note that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1, 2, 1], [6, -1, 0], [-1 , -2, -1]])*matrix([[-1], [2], [1]]) = matrix([[4], [-8], [-4]]);" "6# /*&-%'matrixG6#7%7%\"\"\"\"\"#F*7%\"\"',$F*!\"\"\"\"!7%,$F*F/,$F+F/,$F *F/F*-F&6#7%7#,$F*F/7#F+7#F*F*-F&6#7%7#\"\"%7#,$\"\")F/7#,$F@F/" } {XPPEDIT 18 0 "``=-4*matrix([[-1], [2], [1]])" "6#/%!G,$*&\"\"%\"\"\"- %'matrixG6#7%7#,$F(!\"\"7#\"\"#7#F(F(F/" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 44 "Eigenvectors asso ciated with the eigenvalue " }{XPPEDIT 18 0 "lambda = 3;" "6#/%'lambda G\"\"$" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "Consider the \+ equation " }{XPPEDIT 18 0 "(A-3*I)*`.`*X = ``;" "6#/*(,&%\"AG\"\"\"*& \"\"$F'%\"IGF'!\"\"F'%\".GF'%\"XGF'%!G" }{TEXT 305 1 "0" }{TEXT -1 11 ", that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matri x([[-2, 2, 1], [6, -4, 0], [-1, -2, -4]])*matrix([[x], [y], [z]]) = ma trix([[0], [0], [0]]);" "6#/*&-%'matrixG6#7%7%,$\"\"#!\"\"F+\"\"\"7%\" \"',$\"\"%F,\"\"!7%,$F-F,,$F+F,,$F1F,F--F&6#7%7#%\"xG7#%\"yG7#%\"zGF-- F&6#7%7#F27#F27#F2" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "X=matrix([[ x],[y],[z]])" "6#/%\"XG-%'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 88 "We solve this system of equations by applying row operations to the c oefficient matrix: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[-2, 2, 1], [6, -4, 0], [-1, -2, -4]])" "6#-%'matrixG6#7%7% ,$\"\"#!\"\"F)\"\"\"7%\"\"',$\"\"%F*\"\"!7%,$F+F*,$F)F*,$F/F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 6 " swap " }{XPPEDIT 18 0 "R[1]" "6#&%\"RG6#\"\"\"" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "R[3]" "6#&%\"RG6#\"\"$" }{TEXT -1 7 " ... " } {XPPEDIT 18 0 "matrix([[-1, -2, -4], [6, -4, 0],[-2, 2, 1]])" "6#-%'ma trixG6#7%7%,$\"\"\"!\"\",$\"\"#F*,$\"\"%F*7%\"\"',$F.F*\"\"!7%,$F,F*F, F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "R[1] = -R[1]" "6#/&%\"RG6#\"\"\",$&F% 6#F'!\"\"" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[1, 2, 4], [6 , -4, 0],[-2, 2, 1]])" "6#-%'matrixG6#7%7%\"\"\"\"\"#\"\"%7%\"\"',$F*! \"\"\"\"!7%,$F)F.F)F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([R[2] = R[2]-6*R[1], ``],[R[3] = R[3]+2*R[1], ``]);" "6#-%*PIECEWISEG6$7$/&% \"RG6#\"\"#,&&F)6#F+\"\"\"*&\"\"'F/&F)6#F/F/!\"\"%!G7$/&F)6#\"\"$,&&F) 6#F:F/*&F+F/&F)6#F/F/F/F5" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matri x([[1, 2, 4], [0, -16, -24],[0, 6, 9]])" "6#-%'matrixG6#7%7%\"\"\"\"\" #\"\"%7%\"\"!,$\"#;!\"\",$\"#CF/7%F,\"\"'\"\"*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([R[2] = R[2]*`.`*``(-1/16), ``],[R[2] = R[2]* `.`*``(1/6), ``]);" "6#-%*PIECEWISEG6$7$/&%\"RG6#\"\"#*(&F)6#F+\"\"\"% \".GF/-%!G6#,$*&F/F/\"#;!\"\"F7F/F27$/&F)6#F+*(&F)6#F+F/F0F/-F26#*&F/F /\"\"'F7F/F2" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[1, 2, 4], [0, 1, 3/2],[0, 1, 3/2]])" "6#-%'matrixG6#7%7%\"\"\"\"\"#\"\"%7%\"\"! F(*&\"\"$F(F)!\"\"7%F,F(*&F.F(F)F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECE WISE(([R[1] = R[1]-2*R[2],``],[R[3] = R[3]-R[2],``])" "6#-%*PIECEWISEG 6$7$/&%\"RG6#\"\"\",&&F)6#F+F+*&\"\"#F+&F)6#F0F+!\"\"%!G7$/&F)6#\"\"$, &&F)6#F9F+&F)6#F0F3F4" }{TEXT -1 7 " ... " }{XPPEDIT 18 0 "matrix([[ 1, 0, 1], [0, 1, 3/2],[0, 0,0]])" "6#-%'matrixG6#7%7%\"\"\"\"\"!F(7%F) F(*&\"\"$F(\"\"#!\"\"7%F)F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "PIECEWISE([x+z = 0, ``],[y+3/2*z = 0, ``]);" "6 #-%*PIECEWISEG6$7$/,&%\"xG\"\"\"%\"zGF*\"\"!%!G7$/,&%\"yGF**(\"\"$F*\" \"#!\"\"F+F*F*F,F-" }{TEXT -1 7 " or " }{XPPEDIT 18 0 "PIECEWISE([x = -z, ``],[y = -3/2*z, ``]);" "6#-%*PIECEWISEG6$7$/%\"xG,$%\"zG!\"\"% !G7$/%\"yG,$*(\"\"$\"\"\"\"\"#F+F*F3F+F," }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "z = -2*t;" "6#/%\"zG,$*&\"\"# \"\"\"%\"tGF(!\"\"" }{TEXT -1 7 ". Then " }{XPPEDIT 18 0 "x = 2*t;" "6 #/%\"xG*&\"\"#\"\"\"%\"tGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 3 *t;" "6#/%\"yG*&\"\"$\"\"\"%\"tGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 71 "There are infinitely many solutions given by the parame tric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "P IECEWISE([x = 2*t, ``],[y = 3*t, ``],[z = -2*t, ``]);" "6#-%*PIECEWISE G6%7$/%\"xG*&\"\"#\"\"\"%\"tGF+%!G7$/%\"yG*&\"\"$F+F,F+F-7$/%\"zG,$*&F *F+F,F+!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 72 "These \+ equations can be written in the form of a single vector equation: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x], [y], [z] ]) = t*matrix([[2], [3], [-2]]);" "6#/-%'matrixG6#7%7#%\"xG7#%\"yG7#% \"zG*&%\"tG\"\"\"-F%6#7%7#\"\"#7#\"\"$7#,$F5!\"\"F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Such vectors are eigenvectors provided \+ that " }{XPPEDIT 18 0 "t <>0" "6#0%\"tG\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 22 "As a check note that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1, 2, 1], [6, -1, 0], [-1, \+ -2, -1]])*matrix([[2], [3], [-2]]) = matrix([[6], [9], [-6]]);" "6#/*& -%'matrixG6#7%7%\"\"\"\"\"#F*7%\"\"',$F*!\"\"\"\"!7%,$F*F/,$F+F/,$F*F/ F*-F&6#7%7#F+7#\"\"$7#,$F+F/F*-F&6#7%7#F-7#\"\"*7#,$F-F/" }{XPPEDIT 18 0 "`` = 3*matrix([[2], [3], [-2]]);" "6#/%!G*&\"\"$\"\"\"-%'matrixG 6#7%7#\"\"#7#F&7#,$F-!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 10 "Conclusion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 56 "The eigenvalues and their associat ed eigenvectors are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda = 0,t*matrix([[1], [6], [-13]]),t <> 0;" "6%/%'lambdaG\" \"!*&%\"tG\"\"\"-%'matrixG6#7%7#F(7#\"\"'7#,$\"#8!\"\"F(0F'F%" }{TEXT -1 5 "; " }{XPPEDIT 18 0 "lambda = -4,t*matrix([[-1], [2], [1]]),t \+ <> 0;" "6%/%'lambdaG,$\"\"%!\"\"*&%\"tG\"\"\"-%'matrixG6#7%7#,$F*F'7# \"\"#7#F*F*0F)\"\"!" }{TEXT -1 6 "; " }{XPPEDIT 18 0 "lambda = 3,t *matrix([[2], [3], [-2]]),t <> 0;" "6%/%'lambdaG\"\"$*&%\"tG\"\"\"-%'m atrixG6#7%7#\"\"#7#F%7#,$F.!\"\"F(0F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "A \+ := matrix([[1,2,1],[6,-1,0],[-1,-2,-1]]);\nlinalg[eigenvalues](A);\nli nalg[eigenvectors](A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'mat rixG6#7%7%\"\"\"\"\"#F*7%\"\"'!\"\"\"\"!7%F.!\"#F." }}{PARA 11 "" 1 " " {XPPMATH 20 "6%\"\"!\"\"$!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%! \"%\"\"\"<#-%'vectorG6#7%!\"\"\"\"#F%7%\"\"$F%<#-F(6#7%F+#!\"$F,F%7%\" \"!F%<#-F(6#7%F%\"\"'!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "A := Matrix([[1,2,1],[6,-1,0],[-1,- 2,-1]]);\nLinearAlgebra[Eigenvectors](A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*s!4_9-%'MATRIXG6#7%7%\"\"\"\"\"#F. 7%\"\"'!\"\"\"\"!7%F2!\"#F2%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$-%'RTABLEG6%\"*c\\,X\"-%'MATRIXG6#7%7#\"\"!7#!\"%7#\"\"$&%'VectorG6# %'columnG-F$6%\"*#f:_9-F(6#7%7%\"\"\"!\"\"F=7%\"\"'\"\"##!\"$F@7%!#8F< F<%'MatrixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 126 "In questions 1 to 5 find the characteristic equat ion of the given matrix A. Also find the eigenvalues and the eigenvect ors of " }{TEXT 306 1 "A" }{TEXT -1 2 ". " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A=ma trix([[-1,2],[-7,8]])" "6#/%\"AG-%'matrixG6#7$7$,$\"\"\"!\"\"\"\"#7$,$ \"\"(F,\"\")" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 25 "characteristic equation " }{XPPEDIT 18 0 "lambda^2-7*lambda+6=0" "6#/,(*$%'lambdaG\"\"#\"\"\"*&\"\"(F(F&F(!\" \"\"\"'F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "lambda=1" "6#/%'lambdaG\"\"\"" }{TEXT -1 16 ", eigenvec tors " }{XPPEDIT 18 0 " t*matrix([[1],[1]]), t<>0" "6$*&%\"tG\"\"\"-% 'matrixG6#7$7#F%7#F%F%0F$\"\"!" }{TEXT -1 3 "; " }{XPPEDIT 18 0 "lamb da=6" "6#/%'lambdaG\"\"'" }{TEXT -1 16 ", eigenvectors " }{XPPEDIT 18 0 "t*matrix([[2],[7]]), t<>0" "6$*&%\"tG\"\"\"-%'matrixG6#7$7#\"\"# 7#\"\"(F%0F$\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "A := Matrix([[-1,2],[-7,8]] );\nLinearAlgebra[CharacteristicMatrix](A,lambda);\nLinearAlgebra[Dete rminant](%);\nLinearAlgebra[Eigenvectors](A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*Wq@X\"-%'MATRIXG6#7$7$!\"\"\"\"#7$ !\"(\"\")%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"*k 8AX\"-%'MATRIXG6#7$7$,&%'lambdaG!\"\"\"\"\"F.\"\"#7$!\"(,&F-F.\"\")F/% 'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%'lambdaG\"\"#\"\"\"F (*&\"\"(F(F&F(!\"\"\"\"'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABL EG6%\"*c^,X\"-%'MATRIXG6#7$7#\"\"\"7#\"\"'&%'VectorG6#%'columnG-F$6%\" *?aAX\"-F(6#7$7$F,F,7$F,#\"\"(\"\"#%'MatrixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________ ____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = matrix([[4, 8], [0, -5]]);" "6#/%\"AG-%'matrix G6#7$7$\"\"%\"\")7$\"\"!,$\"\"&!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "characteristic eq uation " }{XPPEDIT 18 0 "lambda^2+lambda-20=0" "6#/,(*$%'lambdaG\"\"# \"\"\"F&F(\"#?!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda = -5;" "6#/%'lambdaG,$\"\"&!\"\"" }{TEXT -1 16 ", eigenvectors " }{XPPEDIT 18 0 "t*matrix([[8], [-9]]),t <> 0; " "6$*&%\"tG\"\"\"-%'matrixG6#7$7#\"\")7#,$\"\"*!\"\"F%0F$\"\"!" } {TEXT -1 3 "; " }{XPPEDIT 18 0 "lambda = 4;" "6#/%'lambdaG\"\"%" } {TEXT -1 16 ", eigenvectors " }{XPPEDIT 18 0 "t*matrix([[1], [0]]),t \+ <> 0;" "6$*&%\"tG\"\"\"-%'matrixG6#7$7#F%7#\"\"!F%0F$F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "A := Matrix([[4, 8], [0, -5]]);\nLinearAlgebra[Chara cteristicMatrix](A,lambda);\nexpand(LinearAlgebra[Determinant](%));\nf actor(%);\nLinearAlgebra[Eigenvectors](A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*'*oAX\"-%'MATRIXG6#7$7$\"\"%\"\")7 $\"\"!!\"&%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"* )=K_9-%'MATRIXG6#7$7$,&%'lambdaG!\"\"\"\"%\"\"\"\"\")7$\"\"!,&F-F.\"\" &F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%'lambdaG\"\"#\" \"\"F(F&F(\"#?!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%'lambdaG\" \"\"\"\"&F&F&,&F%F&\"\"%!\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%' RTABLEG6%\"*c`,X\"-%'MATRIXG6#7$7#!\"&7#\"\"%&%'VectorG6#%'columnG-F$6 %\"*)QO_9-F(6#7$7$\"\"\"F:7$#!\"*\"\")\"\"!%'MatrixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "___________ ________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = matrix([[5,-1,0],[0,-5,9],[5,-1,0]])" "6#/%\"A G-%'matrixG6#7%7%\"\"&,$\"\"\"!\"\"\"\"!7%F.,$F*F-\"\"*7%F*,$F,F-F." } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 25 "characteristic equation " }{XPPEDIT 18 0 "lambda^3-16* lambda=0" "6#/,&*$%'lambdaG\"\"$\"\"\"*&\"#;F(F&F(!\"\"\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda = 0; " "6#/%'lambdaG\"\"!" }{TEXT -1 16 ", eigenvectors " }{XPPEDIT 18 0 " t*matrix([[9], [45], [25]]),t <> 0;" "6$*&%\"tG\"\"\"-%'matrixG6#7%7# \"\"*7#\"#X7#\"#DF%0F$\"\"!" }{TEXT -1 3 "; " }{XPPEDIT 18 0 "lambda \+ = 4;" "6#/%'lambdaG\"\"%" }{TEXT -1 16 ", eigenvectors " }{XPPEDIT 18 0 "t*matrix([[1], [1], [1]]),t <> 0;" "6$*&%\"tG\"\"\"-%'matrixG6#7 %7#F%7#F%7#F%F%0F$\"\"!" }{TEXT -1 3 "; " }{XPPEDIT 18 0 "lambda = -4 ;" "6#/%'lambdaG,$\"\"%!\"\"" }{TEXT -1 16 ", eigenvectors " } {XPPEDIT 18 0 "t*matrix([[1], [9], [1]]),t <> 0;" "6$*&%\"tG\"\"\"-%'m atrixG6#7%7#F%7#\"\"*7#F%F%0F$\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "A := Matr ix([[5,-1,0],[0,-5,9],[5,-1,0]]);\nLinearAlgebra[CharacteristicMatrix] (A,lambda);\nexpand(LinearAlgebra[Determinant](%));\nfactor(%);\nLinea rAlgebra[Eigenvectors](A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-% 'RTABLEG6%\"*kyBX\"-%'MATRIXG6#7%7%\"\"&!\"\"\"\"!7%F0!\"&\"\"*F-%'Mat rixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"*%oS_9-%'MATRIXG 6#7%7%,&%'lambdaG!\"\"\"\"&\"\"\"F.\"\"!7%F1,&F-F.F/F.\"\"*7%F/F.,$F-F .%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%'lambdaG\"\"$\"\" \"!\"\"*&\"#;F(F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%'lambdaG \"\"\",&F%F&\"\"%!\"\"F&,&F%F&F(F&F&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"*cb,X\"-%'MATRIXG6#7%7#\"\"%7#!\"%7#\"\"!&%'Vector G6#%'columnG-F$6%\"*_lCX\"-F(6#7%7%\"\"\"F%'MatrixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "A=matrix([[1,2,3] ,[0,5,6],[0,0,-7]])" "6#/%\"AG-%'matrixG6#7%7%\"\"\"\"\"#\"\"$7%\"\"! \"\"&\"\"'7%F.F.,$\"\"(!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "characteristic equation \+ " }{XPPEDIT 18 0 "lambda^3+lambda^2-37*lambda+35=0" "6#/,**$%'lambdaG \"\"$\"\"\"*$F&\"\"#F(*&\"#PF(F&F(!\"\"\"#NF(\"\"!" }{TEXT -1 4 " or \+ " }{XPPEDIT 18 0 "(lambda-5)*(lambda-1)*(lambda+7)=0" "6#/*(,&%'lambda G\"\"\"\"\"&!\"\"F',&F&F'F'F)F',&F&F'\"\"(F'F'\"\"!" }}{PARA 0 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "lambda = 5;" "6#/%'lambdaG\"\"&" } {TEXT -1 16 ", eigenvectors " }{XPPEDIT 18 0 "t*matrix([[1], [2], [0] ]),t <> 0;" "6$*&%\"tG\"\"\"-%'matrixG6#7%7#F%7#\"\"#7#\"\"!F%0F$F." } {TEXT -1 3 "; " }{XPPEDIT 18 0 "lambda = 1;" "6#/%'lambdaG\"\"\"" } {TEXT -1 16 ", eigenvectors " }{XPPEDIT 18 0 "t*matrix([[1], [0], [0] ]),t <> 0;" "6$*&%\"tG\"\"\"-%'matrixG6#7%7#F%7#\"\"!7#F,F%0F$F," } {TEXT -1 3 "; " }{XPPEDIT 18 0 "lambda = -7;" "6#/%'lambdaG,$\"\"(!\" \"" }{TEXT -1 16 ", eigenvectors " }{XPPEDIT 18 0 "t*matrix([[1], [2] , [-4]]),t <> 0;" "6$*&%\"tG\"\"\"-%'matrixG6#7%7#F%7#\"\"#7#,$\"\"%! \"\"F%0F$\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "A := Matrix([[1,2,3],[0,5,6 ],[0,0,-7]]);\nLinearAlgebra[CharacteristicMatrix](A,lambda);\nexpand( LinearAlgebra[Determinant](%));\nfactor(%);\nLinearAlgebra[Eigenvector s](A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*/![_9-% 'MATRIXG6#7%7%\"\"\"\"\"#\"\"$7%\"\"!\"\"&\"\"'7%F2F2!\"(%'MatrixG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"*WLDX\"-%'MATRIXG6#7%7%, &%'lambdaG!\"\"\"\"\"F/\"\"#\"\"$7%\"\"!,&F-F.\"\"&F/\"\"'7%F3F3,&F-F. \"\"(F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%'lambdaG\"\" $\"\"\"!\"\"*$)F&\"\"#F(F)*&\"#PF(F&F(F(\"#NF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%'lambdaG\"\"\"\"\"&!\"\"F',&F&F'F'F)F',&F&F'\"\" (F'F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"*cd,X\"-%'MAT RIXG6#7%7#!\"(7#\"\"&7#\"\"\"&%'VectorG6#%'columnG-F$6%\"*O&f_9-F(6#7% 7%F0F0F07%\"\"#F=\"\"!7%!\"%F>F>%'MatrixG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "__________________ _________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "A = matrix([[-1,1,0],[1,2,1],[0,3,-1]])" "6#/%\"AG-%'ma trixG6#7%7%,$\"\"\"!\"\"F+\"\"!7%F+\"\"#F+7%F-\"\"$,$F+F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "characteristic equation " }{XPPEDIT 18 0 "lambda^3-7*lambda-6= 0" "6#/,(*$%'lambdaG\"\"$\"\"\"*&\"\"(F(F&F(!\"\"\"\"'F+\"\"!" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "(lambda-3)*(lambda+2)*(lambda+1)=0" "6#/ *(,&%'lambdaG\"\"\"\"\"$!\"\"F',&F&F'\"\"#F'F',&F&F'F'F'F'\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "lambd a = 3;" "6#/%'lambdaG\"\"$" }{TEXT -1 16 ", eigenvectors " }{XPPEDIT 18 0 "t*matrix([[1], [4], [3]]),t <> 0;" "6$*&%\"tG\"\"\"-%'matrixG6#7 %7#F%7#\"\"%7#\"\"$F%0F$\"\"!" }{TEXT -1 3 "; " }{XPPEDIT 18 0 "lambd a = -1;" "6#/%'lambdaG,$\"\"\"!\"\"" }{TEXT -1 16 ", eigenvectors " } {XPPEDIT 18 0 "t*matrix([[-1], [0], [1]]),t <> 0;" "6$*&%\"tG\"\"\"-%' matrixG6#7%7#,$F%!\"\"7#\"\"!7#F%F%0F$F." }{TEXT -1 3 "; " }{XPPEDIT 18 0 "lambda = -2;" "6#/%'lambdaG,$\"\"#!\"\"" }{TEXT -1 16 ", eigenve ctors " }{XPPEDIT 18 0 "t*matrix([[1], [-1], [3]]),t <> 0;" "6$*&%\"t G\"\"\"-%'matrixG6#7%7#F%7#,$F%!\"\"7#\"\"$F%0F$\"\"!" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "A := Matrix([[-1,1,0],[1,2,1],[0,3,-1]]);\nLinearAlgebra[Char acteristicMatrix](A,lambda);\nexpand(LinearAlgebra[Determinant](%));\n factor(%);\nLinearAlgebra[Eigenvectors](A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\"*))4EX\"-%'MATRIXG6#7%7%!\"\"\"\"\" \"\"!7%F/\"\"#F/7%F0\"\"$F.%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'RTABLEG6%\"*3QEX\"-%'MATRIXG6#7%7%,&%'lambdaG!\"\"\"\"\"F.F/\"\"! 7%F/,&F-F.\"\"#F/F/7%F0\"\"$F,%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%'lambdaG\"\"$\"\"\"!\"\"*&\"\"(F(F&F(F(\"\"'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,&%'lambdaG\"\"\"\"\"$!\"\"F',&F&F'\"\"# F'F',&F'F'F&F'F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"*c f,X\"-%'MATRIXG6#7%7#!\"\"7#!\"#7#\"\"$&%'VectorG6#%'columnG-F$6%\"*%3 n_9-F(6#7%7%F,F,\"\"\"7%\"\"!F<\"\"%7%F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "________ ___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 38 "numeri cal example of 3 x 3 determinant" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1552 "A := matrix([[2, 4, 1], [1 , 3, 2], [1, -3, -7]]);\np1 := plot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2], [2.4,2.1]]],color=black):\nt1 := plots[textplot]([[0,2,A[1,1]],[1,2,A[ 1,2]],[2,2,A[1,3]],\n [0,1,A[2,1]],[1,1,A[2,2]],[2,1,A[2,3]],\n [0,0 ,A[3,1]],[1,0,A[3,2]],[2,0,A[3,3]]],font=[TIMES,ROMAN,12],color=black) :\nt2 := plots[textplot]([[3,2,A[1,1]],[4,2,A[1,2]],[3,1,A[2,1]],[4,1, A[2,2]],\n [3,0,A[3,1]],[4,0,A[3,2]]],font=[TIMES,ROMAN,12],color=COLO R(RGB,0,.6,0)):\nt3 := plots[textplot]([[3,3,A[3,1]*A[2,2]*A[1,3]],[4, 3,A[3,2]*A[2,3]*A[1,1]],\n [5,3,A[3,3]* A[2,1]*A[1,2]]],\n font=[TIM ES,ROMAN,12],color=COLOR(RGB,0,0,.9)):\nt4 := plots[textplot]([[3,-1,A [1,1]*A[2,2]*A[3,3]],[4,-1,A[1,2]*A[2,3]*A[3,1]],\n [5,-1,A[1,3]*A[2, 1]*A[3,2]]],\n font=[TIMES,ROMAN,12],color=COLOR(RGB,.95,0,0)):\np2 : = plottools[arrow]([-.01,-.01],[2.7,2.7],0,.15,.05,arrow,\n color= COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np3 := plottools[arrow]([ .99,-.01],[3.7,2.7],0,.15,.05,arrow,\n color=COLOR(RGB,.6,.6,1),th ickness=2,linestyle=2):\np4 := plottools[arrow]([1.99,-.01],[4.7,2.7], 0,.15,.05,arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2,linestyle= 2):\np5 := plottools[arrow]([-.01,1.99],[2.7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\np6 := plottoo ls[arrow]([.99,1.99],[3.7,-.702],0,.15,.05,arrow,\n color=COLOR(RG B,1,.3,.3),thickness=2,linestyle=2):\np7 := plottools[arrow]([1.99,1.9 9],[4.7,-.702],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickne ss=2,linestyle=2):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,t1,t2,t3,t4], axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 12 "Examp le 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 454 "AA := matrix([[3, 2, -2], [-1, -4, 1], [2, -4, -1]]) ;\nA := evalm(AA-matrix([[l,0,0], [0,l,0],[0,0,l]]));\np1 := plot([[[- .4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1]]],color=black):\nt1 := plots[t extplot]([[0,2,A[1,1]],[1,2,A[1,2]],[2,2,A[1,3]],\n [0,1,A[2,1]],[1, 1,A[2,2]],[2,1,A[2,3]],\n [0,0,A[3,1]],[1,0,A[3,2]],[2,0,A[3,3]]],fon t=[SYMBOL,12],color=black):\nt2 := plots[textplot]([2.8,1,`= 0.`],font =[TIMES,ROMAN,12]): \nplots[display]([p1,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1268 "AA := matrix([[3, 2, -2], [-1, -4 , 1], [2, -4, -1]]);\nA := evalm(AA-matrix([[l,0,0], [0,l,0],[0,0,l]]) );\np1 := plot([[[-.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1]]],color=bla ck):\nt1 := plots[textplot]([[0,2,A[1,1]],[1,2,A[1,2]],[2,2,A[1,3]],\n [0,1,A[2,1]],[1,1,A[2,2]],[2,1,A[2,3]],\n [0,0,A[3,1]],[1,0,A[3,2]] ,[2,0,A[3,3]]],font=[SYMBOL,12],color=black):\nt2 := plots[textplot]([ [3,2,A[1,1]],[4,2,A[1,2]],[3,1,A[2,1]],[4,1,A[2,2]],\n [3,0,A[3,1]],[4 ,0,A[3,2]]],font=[SYMBOL,12],color=COLOR(RGB,0,.6,0)):\np2 := plottool s[arrow]([-.01,-.01],[2.3,2.3],0,.12,.05,arrow,\n color=COLOR(RGB, .6,.6,1),thickness=2,linestyle=2):\np3 := plottools[arrow]([.99,-.01], [3.3,2.3],0,.12,.05,arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2, linestyle=2):\np4 := plottools[arrow]([1.99,-.01],[4.3,2.3],0,.15,.05, arrow,\n color=COLOR(RGB,.6,.6,1),thickness=2,linestyle=2):\np5 := plottools[arrow]([-.01,1.99],[2.3,-.302],0,.15,.05,arrow,\n color =COLOR(RGB,1,.3,.3),thickness=2,linestyle=2):\np6 := plottools[arrow]( [.99,1.99],[3.3,-.302],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3) ,thickness=2,linestyle=2):\np7 := plottools[arrow]([1.99,1.99],[4.3,-. 302],0,.15,.05,arrow,\n color=COLOR(RGB,1,.3,.3),thickness=2,lines tyle=2):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,t1,t2],axes=none);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "p1 := \+ plot([[[2.5,-.2],[2.5,1.2]],[[4.5,-.2],[4.5,1.2]]],color=black):\nt1 : = plots[textplot]([[3,1,`21 - l`],[4,1,-60],[3,0,8],[4,0,`-23 - l`],\n [4.85,.5,`= 0.`]],font=[SYMBOL,12],color=black):\nplots[display]([p1, t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 12 "E xample 3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 453 "AA := matrix([[1, 2, 1], [6, -1, 0], [-1, -2, -1]] );\nA := evalm(AA-matrix([[l,0,0], [0,l,0],[0,0,l]]));\np1 := plot([[[ -.4,-.2],[-.4,2.1]],[[2.4,-.2],[2.4,2.1]]],color=black):\nt1 := plots[ textplot]([[0,2,A[1,1]],[1,2,A[1,2]],[2,2,A[1,3]],\n [0,1,A[2,1]],[1 ,1,A[2,2]],[2,1,A[2,3]],\n [0,0,A[3,1]],[1,0,A[3,2]],[2,0,A[3,3]]],fo nt=[SYMBOL,12],color=black):\nt2 := plots[textplot]([2.8,1,`= 0.`],fon t=[TIMES,ROMAN,12]): \nplots[display]([p1,t1,t2],axes=none);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 390 "p1 := plot([[[4.1,.3],[4.1,1.6]],[[5.9,.3],[5.9,1.6]],\n [[7.4 ,.3],[7.4,1.6]],[[9.2,.3],[9.2,1.6]]],color=black):\nt1 := plots[textp lot]([[4.5,1.5,`2`],[5.5,1.5,`1`],[4.5,.5,`-2`],[5.4,.5,`-1 - l`],\n \+ [7.8,1.5,`1 - l`],[8.8,1.5,`1`],[7.8,.5,`-1`],[8.7,.5,`-1 - l`],\n [3 .7,1,`( -6 )`],[6.7,1,`+ ( -1 - l )`],[9.5,1,`= 0.`]],font=[SYMBOL,12] ,color=black):\nplots[display]([p1,t1],axes=none);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 145191432 145014636 145198452 145199976 145014796 145206372 145209072 145014956 145215592 145217044 145221364 145015156 145225420 145226896 145232188 145015356 145236388 145237864 145240684 145015556 145246552 145248004 145253344 145015756 145259536 145260988 145263808 145015956 145267084 }{RTABLE M7R0 I6RTABLE_SAVE/145191432X,%)anythingG6"6"[gl!"%!!!#*"$"$""$!""""#F)!"%F*!"#"""F( 6" } {RTABLE M7R0 I6RTABLE_SAVE/145014636X*%*algebraicG6"6"[gl!#%!!!"$"$!"#"""!""6" } {RTABLE M7R0 I6RTABLE_SAVE/145198452X,%*algebraicG6"6"[gl!"%!!!#*"$"$"""#""$""#""%F'""!F'F*F '""&6" } {RTABLE M7R0 I6RTABLE_SAVE/145199976X,%)anythingG6"6"[gl!"%!!!#%"#"#"#@"")!#g!#B6" } {RTABLE M7R0 I6RTABLE_SAVE/145014796X*%*algebraicG6"6"[gl!#%!!!"#"#!"$"""6" } {RTABLE M7R0 I6RTABLE_SAVE/145206372X,%*algebraicG6"6"[gl!"%!!!#%"#"##""&""#"""""$F*6" } {RTABLE M7R0 I6RTABLE_SAVE/145209072X,%)anythingG6"6"[gl!"%!!!#*"$"$"""""'!""""#F)!"#F'""!F) 6" } {RTABLE M7R0 I6RTABLE_SAVE/145014956X*%*algebraicG6"6"[gl!#%!!!"$"$""!!"%""$6" } {RTABLE M7R0 I6RTABLE_SAVE/145215592X,%*algebraicG6"6"[gl!"%!!!#*"$"$"""""'!#8!""""#F'F*#!"$ F+F'6" } {RTABLE M7R0 I6RTABLE_SAVE/145217044X,%)anythingG6"6"[gl!"%!!!#%"#"#!""!"(""#"")6" } {RTABLE M7R0 I6RTABLE_SAVE/145221364X,%)anythingG6"6"[gl!"%!!!#%"#"#,&%'lambdaG!""F)"""!"("" #,&F(F)"")F*6" } {RTABLE M7R0 I6RTABLE_SAVE/145015156X*%*algebraicG6"6"[gl!#%!!!"#"#"""""'6" } {RTABLE M7R0 I6RTABLE_SAVE/145225420X,%*algebraicG6"6"[gl!"%!!!#%"#"#"""F'F'#""(""#6" } {RTABLE M7R0 I6RTABLE_SAVE/145226896X,%)anythingG6"6"[gl!"%!!!#%"#"#""%""!"")!"&6" } {RTABLE M7R0 I6RTABLE_SAVE/145232188X,%)anythingG6"6"[gl!"%!!!#%"#"#,&%'lambdaG!""""%"""""!" "),&F(F)!"&F+6" } {RTABLE M7R0 I6RTABLE_SAVE/145015356X*%*algebraicG6"6"[gl!#%!!!"#"#!"&""%6" } {RTABLE M7R0 I6RTABLE_SAVE/145236388X,%*algebraicG6"6"[gl!"%!!!#%"#"#"""#!"*"")F'""!6" } {RTABLE M7R0 I6RTABLE_SAVE/145237864X,%)anythingG6"6"[gl!"%!!!#*"$"$""&""!F'!""!"&F)F(""*F(6 " } {RTABLE M7R0 I6RTABLE_SAVE/145240684X,%)anythingG6"6"[gl!"%!!!#*"$"$,&%'lambdaG!""""&"""""!F *F),&F(F)!"&F+F)F,""*,$F(F)6" } {RTABLE M7R0 I6RTABLE_SAVE/145015556X*%*algebraicG6"6"[gl!#%!!!"$"$""%!"%""!6" } {RTABLE M7R0 I6RTABLE_SAVE/145246552X,%*algebraicG6"6"[gl!"%!!!#*"$"$"""F'F'F'""*F'F'""&#"#D F(6" } {RTABLE M7R0 I6RTABLE_SAVE/145248004X,%)anythingG6"6"[gl!"%!!!#*"$"$"""""!F(""#""&F(""$""'!" (6" } {RTABLE M7R0 I6RTABLE_SAVE/145253344X,%)anythingG6"6"[gl!"%!!!#*"$"$,&%'lambdaG!"""""F*""!F+ ""#,&F(F)""&F*F+""$""',&F(F)!"(F*6" } {RTABLE M7R0 I6RTABLE_SAVE/145015756X*%*algebraicG6"6"[gl!#%!!!"$"$!"(""&"""6" } {RTABLE M7R0 I6RTABLE_SAVE/145259536X,%*algebraicG6"6"[gl!"%!!!#*"$"$"""""#!"%F'F(""!F'F*F*6 " } {RTABLE M7R0 I6RTABLE_SAVE/145260988X,%)anythingG6"6"[gl!"%!!!#*"$"$!"""""""!F(""#""$F)F(F'6 " } {RTABLE M7R0 I6RTABLE_SAVE/145263808X,%)anythingG6"6"[gl!"%!!!#*"$"$,&%'lambdaG!""F)"""F*""! F*,&F(F)""#F*""$F+F*F'6" } {RTABLE M7R0 I6RTABLE_SAVE/145015956X*%*algebraicG6"6"[gl!#%!!!"$"$!""!"#""$6" } {RTABLE M7R0 I6RTABLE_SAVE/145267084X,%*algebraicG6"6"[gl!"%!!!#*"$"$!""""!"""F'F)!"$F)""%"" $6" }