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{SECT 0 {PARA 4 "" 0 "" {TEXT 228 25 "Linear Algebra with Maple" }} {PARA 0 "" 0 "" {TEXT 229 0 "" }}{PARA 0 "" 0 "" {TEXT 207 52 "Please \+ execute the next two lines before proceeding:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 20 "with(LinearAlgebra): " }}}{PARA 0 "" 0 "" {TEXT 207 0 "" }}{PARA 0 "" 0 "" {TEXT 207 82 "Ma ple can be used to solve problems from Linear Algebra. It is possible \+ to define " }{TEXT 210 1 "N" }{TEXT 207 25 "-dimensionial vectors and" }{TEXT 211 4 " NxN" }{TEXT 207 88 " matrices and to calculate determi nants, matrix inverses, and solve eigenvalue problems." }}{PARA 0 "" 0 "" {TEXT 207 78 "For example, let us look at a system of three linear equations with variables " }{TEXT 212 1 "x" }{TEXT 207 2 ", " }{TEXT 213 1 "y" }{TEXT 207 6 ", and " }{TEXT 214 2 "z " }{TEXT 207 45 "(take n from Section 4 of \"Maple Essentials\")." }}{PARA 0 "" 0 "" {TEXT 207 21 "The 3 by 3 system: " }{XPPEDIT 2 0 "Typesetting:-mrow(Typese tting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"x\", it alic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"+\", mat hvariant = \"normal\", fence = \"false\", separator = \"false\", stret chy = \"false\", symmetric = \"false\", largeop = \"false\", movableli mits = \"false\", accent = \"false\", lspace = \"0.2222222em\", 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etting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi( \"\"), Typesetting:-mrow(Typesetting:-mn(\"3\", mathvariant = \"normal \"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", f ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi (\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mo (\"+\", mathvariant = \"normal\", fence = \"false\", separator = \"fal se\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\" , movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222e m\", rspace = \"0.2222222em\"), Typesetting:-mi(\"y\", italic = \"true \", mathvariant = \"italic\")), Typesetting:-mo(\"=\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = 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movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"y \", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mo(\"& minus;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.222222 2em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"z\", italic = \"tr ue\", mathvariant = \"italic\")), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fa lse\", symmetric = \"false\", largeop = \"false\", movablelimits = \"f alse\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.277 7778em\"), Typesetting:-mn(\"0\", mathvariant = \"normal\")), Typesett ing:-mi(\"\")), mathvariant = \"normal\", open = \"\{\", close = \"\} \"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I(mfen cedGF$6&-F#6)-I#miGF$6#Q!F'-F#6&F0-F#6'-F16%Q\"xF'/%'italicGQ%trueF'/% ,mathvariantGQ'italicF'-I#moGF$6-Q\"+F'/F?Q'normalF'/%&fenceGQ&falseF' /%*separatorGFI/%)stretchyGFI/%*symmetricGFI/%(largeopGFI/%.movablelim itsGFI/%'accentGFI/%'lspaceGQ,0.2222222emF'/%'rspaceGFX-F16%Q\"yF'F;F> FA-F16%Q\"zF'F;F>-FB6-Q\"=F'FEFGFJFLFNFPFRFT/FWQ,0.2777778emF'/FZF_o-I #mnGF$6$Q\"1F'FE-FB6-Q\",F'FEFG/FKF=FLFNFPFRFT/FWQ&0.0emF'/FZQ,0.33333 33emF'-F#6&F0-F#6&F0-F#6%-Fbo6$Q\"3F'FE-FB6-Q1⁢F'FEFGFJ FLFNFPFRFTFio/FZFjoF8FAFenF[oFcpFeo-F#6&F0-F#6'F8-FB6-Q(−F'FEFGF JFLFNFPFRFTFVFY-F#6%-Fbo6$Q\"2F'FEFfpFenF^qFhnF[o-Fbo6$Q\"0F'FEF0FE/%% openGQ\"|frF'/%&closeGQ\"|hrF'" }{TEXT 207 58 " can be easily solved u sing the universal \"solve\" command:" }}{PARA 0 "" 0 "" {TEXT 207 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "solve(\{x+y+z=1, 3*x+y=3 , x-2*y-z=0\});" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "<% /I\"xG6\"#\"\"%\"\"&/I\"yGF%#\"\"$F(/I\"zGF%#!\"#F(" }}}{PARA 0 "" 0 " " {TEXT 207 0 "" }}{PARA 0 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\"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"column\", italic = \"false\", \+ mathvariant = \"normal\"), Typesetting:-mo(\" \", mathvariant = \"norm al\", fence = \"false\", separator = \"false\", stretchy = \"false\", \+ symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesett ing:-mi(\"vector\", italic = \"false\", mathvariant = \"normal\"), Typ esetting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", separ ator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.0em\", rspace = \"0.0em\"), Typesetting:-mo(\" \", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Ty pesetting:-mi(\"a\", italic = \"false\", mathvariant = \"normal\"), Ty pesetting:-mo(\" \", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mo(\"=\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777 778em\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"f alse\", separator = \"false\", stretchy = \"false\", symmetric = \"fal se\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-maction(Type setting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting: -mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mrow(Typesettin g:-mn(\"1\", foreground = \"[200,0,200]\", mathvariant = \"normal\")), rowalign = \"\", columnalign = \"\", groupalign = \"\", rowspan = \"1 \", columnspan = \"1\"), rowalign = \"\", columnalign = \"\", groupali gn = \"\"), Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mrow(Typese tting:-mn(\"3\", foreground = \"[200,0,200]\", mathvariant = \"normal \")), rowalign = \"\", columnalign = \"\", groupalign = \"\", rowspan \+ = \"1\", columnspan = \"1\"), rowalign = \"\", columnalign = \"\", gro upalign = \"\"), Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mrow(T ypesetting:-mn(\"0\", foreground = \"[200,0,200]\", mathvariant = \"no rmal\")), rowalign = \"\", columnalign = \"\", groupalign = \"\", rows pan = \"1\", columnspan = \"1\"), rowalign = \"\", columnalign = \"\", groupalign = \"\"), align = \"axis\", rowalign = \"baseline\", column align = \"center\", groupalign = \"\{left\}\", alignmentscope = \"true \", columnwidth = \"auto\", width = \"auto\", rowspacing = \"1.0ex\", \+ columnspacing = \"0.8em\", rowlines = \"none\", columnlines = \"none\" , frame = \"none\", framespacing = \"0.4em 0.5ex\", equalrows = \"fals e\", equalcolumns = \"false\", displaystyle = \"false\", side = \"righ t\", minlabelspacing = \"0.8em\"), Typesetting:-mi(\"\")), mathvariant = \"normal\", Typesetting:-msemantics = \"ColVector\", open = \"[\", \+ close = \"]\", Typesetting:-msemantics = \"ColVector\"), actiontype = \+ \"rtableaddress\"), Typesetting:-mo(\".\", mathvariant = \"normal\", f ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"));" "-I%mrowG6#/I +modulenameG6\"I,TypesettingGI(_syslibGF'65-I#miGF$6#Q!F'-I(mactionGF$ 6$-I(mfencedGF$6(-F#6%F+-I'mtableGF$67-I$mtrGF$6(-I$mtdGF$6(-F#6#-I#mn GF$6%Q\"1F'/%+foregroundGQ,[200,0,200]F'/%,mathvariantGQ'normalF'/%)ro walignGF./%,columnalignGF./%+groupalignGF./%(rowspanGQ\"1F'/%+columnsp anGFTF=F=FLFNFP-F;6(-F>6(-F#6#-FC6%Q\"3F'FFFIFLFNFPFRFUF=-F>6(-F#6#-FC 6%Q\"0F'FFFIFLFNFPFRFUFLFNFP-F;6(F=-F>6(-F#6$-I#moGF$6.Q*&uminus0;F'FF FI/%&fenceGQ&falseF'/%*separatorGF]p/%)stretchyGF]p/%*symmetricGF]p/%( largeopGF]p/%.movablelimitsGF]p/%'accentGF]p/%'lspaceGQ,0.2222222emF'/ %'rspaceGF\\q-FC6%Q\"2F'FFFIFLFNFPFRFU-F>6(-F#6$FgoFBFLFNFPFRFUFLFNFP/ %&alignGQ%axisF'/FMQ)baselineF'/FOQ'centerF'/FQQ'|frleft|hrF'/%/alignm entscopeGQ%trueF'/%,columnwidthGQ%autoF'/%&widthGFdr/%+rowspacingGQ&1. 0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF_s/ %&frameGF_s/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGF]p/%-equalcolu mnsGF]p/%-displaystyleGF]p/%%sideGQ&rightF'/%0minlabelspacingGF\\sF+FI /I+msemanticsGF$Q'MatrixF'/%%openGQ\"[F'/%&closeGQ\"]F'Fbt/%+actiontyp eGQ.rtableaddressF'-Fho6-Q\"~F'FIF[pF^pF`pFbpFdpFfpFhp/F[qQ&0.0emF'/F^ qFbu-Fho6.F`u/%'italicGFar/FJQ'italicF'F[pF^pF`pFbpFdpFfpFhpFauFcu-Fho 6.Q$andF'/%%boldGF]pFIF[pF^pF`pFbpFdpFfpFhpFauFcuF^u-F,6%Q$theF'/FguF] pFIF^u-F,6%Q'columnF'FbvFIF^u-F,6%Q'vectorF'FbvFIF^uF^u-F,6%Q\"aF'FbvF IF^u-Fho6-Q\"=F'FIF[pF^pF`pFbpFdpFfpFhp/F[qQ,0.2777778emF'/F^qF`wF^u-F 06$-F36(-F#6%F+-F867-F;6&F=FLFNFP-F;6&FYFLFNFP-F;6&FjnFLFNFPFfqFiqF[rF ]rF_rFbrFerFgrFjrF]sF`sFbsFdsFgsFisF[tF]tF`tF+FI/FctQ*ColVectorF'FetFh tF`xF[u-Fho6-Q\".F'FIF[pF^pF`pFbpFdpFfpFhpFauFcu" }}{PARA 0 "" 0 "" {TEXT 207 40 "These objects can be defined as follows:" }}{PARA 0 "" 0 "" {TEXT 207 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "M := Ma trix(3,3,[[1,1,1],[3,1,0],[1,-2,-1]]);\n" }{MPLTEXT 1 0 21 "a := Vecto r([1,3,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'MatrixG6$%*protectedGI (_syslibG6\"6#/I$%idGF'\")G9B_" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I'Ve ctorG6$%*protectedGI(_syslibG6\"6#I'columnGF(6#/I$%idGF(\")/j^_" }}} {PARA 0 "" 0 "" {TEXT 207 0 "" }}{PARA 0 "" 0 "" {TEXT 207 63 "The sol ution of the matrix equation can be expressed using the " }{TEXT 217 7 "inverse" }{TEXT 207 8 " of m: " }{TEXT 218 4 "v = " }{XPPEDIT 207 0 "Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"M\", bold = \+ \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mrow(Typesetting:-mo(\"&uminus0;\", bold = \+ \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"fal se\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mn (\"1\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\" )), superscriptshift = \"0\"));" "-I%mrowG6#/I+modulenameG6\"I,Typeset tingGI(_syslibGF'6#-I%msupGF$6%-I#miGF$6'Q\"MF'/%%boldGQ%trueF'/%'ital icGF4/%,mathvariantGQ,bold-italicF'/%+fontweightGQ%boldF'-F#6$-I#moGF$ 6/Q*&uminus0;F'F2/F8F " 0 "" {MPLTEXT 1 0 24 "MI := MatrixInverse(M);\n" }{MPLTEXT 1 0 32 "v := MatrixVectorMu ltiply(MI,a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I'MatrixG6$%*protected GI(_syslibG6\"6#/I$%idGF'\")[Uo_" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I' VectorG6$%*protectedGI(_syslibG6\"6#I'columnGF(6#/I$%idGF(\")'p;F&" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 207 106 "These commands are a bit length y, and there is a shorter alternative, using the matrix evaluation com mand " }{TEXT 221 9 "evalm( ) " }{TEXT 207 176 "which simply takes as \+ its argument a valid combination of matrices and vectors and evaluates it. There is one little inconvenience: Matrix multiplication requires the operator " }{TEXT 222 2 "&*" }{TEXT 207 27 " instead of simply wr iting " }{TEXT 223 1 "*" }{TEXT 207 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "v2 := evalm(M^(-1)&*a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"$E\\[l$F&#\"\"%\"\"&\"\"##F'F+F'#!\"#F+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 207 46 "Of course it is also possible to \+ evaluate the " }{TEXT 224 12 "determinant " }{TEXT 207 3 "of " }{TEXT 225 1 "M" }{TEXT 207 17 " or to solve the " }{TEXT 226 18 "eigenvalue \+ problem" }{TEXT 227 2 " " }{XPPEDIT 207 0 "Typesetting:-mrow(Typesett ing:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"M\", bold = \"true\", italic = \"true\", `selection-placeholder` = \"true\", mathvariant = \+ \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"−\", \+ bold = \"true\", `selection-placeholder` = \"true\", mathvariant = \"b old\", fontweight = \"bold\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mo(\" \", bold = \"true\", `s election-placeholder` = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"false\", separator = \"false\", stretchy = \"fals e\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fal se\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typ esetting:-msub(Typesetting:-mi(\"λ\", bold = \"true\", italic = \"false\", `selection-placeholder` = \"true\", mathvariant = \"bold\" , fontweight = \"bold\"), Typesetting:-mrow(Typesetting:-mi(\"i\", bol d = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontwe ight = \"bold\")), subscriptshift = \"0\"), Typesetting:-mo(\" \", bol d = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"&Iop f;\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic \", fontweight = \"bold\")), bold = \"true\", `selection-placeholder` \+ = \"true\", mathvariant = \"bold\", fontweight = \"bold\"), Typesettin g:-msub(Typesetting:-mi(\"x\", bold = \"true\", italic = \"true\", mat hvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mrow( Typesetting:-mi(\"i\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\" \", bol d = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.0em\", rspace = \"0.0em\")), subscriptshift = \"0 \"), Typesetting:-mo(\"=\", bold = \"true\", mathvariant = \"bold\", f ontweight = \"bold\", fence = \"false\", separator = \"false\", stretc hy = \"false\", symmetric = \"false\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace \+ = \"0.2777778em\"), Typesetting:-mn(\"0\", bold = \"true\", mathvarian t = \"bold\", fontweight = \"bold\"), Typesetting:-mo(\" \", bold = \" true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mo(\",\", mathva riant = \"normal\", fence = \"false\", separator = \"true\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.333 3333em\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mo(\" \", m athvariant = \"normal\", fence = \"false\", separator = \"false\", str etchy = \"false\", symmetric = \"false\", largeop = \"false\", movable limits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \+ \"0.0em\"), Typesetting:-mi(\"i\", italic = \"true\", mathvariant = \" italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting: -mn(\"1\", mathvariant = \"normal\"), Typesetting:-mo(\" \", mathvaria nt = \"normal\", fence = \"false\", separator = \"false\", stretchy = \+ \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\" ), Typesetting:-mo(\"..\", mathvariant = \"normal\", fence = \"false\" , separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", ls pace = \"0.2222222em\", rspace = \"0.0em\"), Typesetting:-mo(\".\", ma thvariant = \"normal\", fence = \"false\", separator = \"false\", stre tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \" 0.0em\"), Typesetting:-mo(\" \", mathvariant = \"normal\", fence = \"f alse\", separator = \"false\", stretchy = \"false\", symmetric = \"fal se\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"N\", it alic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'62-I(mfencedGF$6'-F#6(-I#miGF$6(Q\"MF'/% %boldGQ%trueF'/%'italicGF6/%6selection-placeholderGF6/%,mathvariantGQ, bold-italicF'/%+fontweightGQ%boldF'-I#moGF$60Q(−F'F4F9/F/%& fenceGQ&falseF'/%*separatorGFH/%)stretchyGFH/%*symmetricGFH/%(largeopG FH/%.movablelimitsGFH/%'accentGFH/%'lspaceGQ,0.2222222emF'/%'rspaceGFW -FB60Q\"~F'F4F9FEF>FFFIFKFMFOFQFS/FVQ&0.0emF'/FYFhn-I%msubGF$6%-F16(Q) λF'F4/F8FHF9FEF>-F#6#-F16'Q\"iF'F4F7F;F>/%/subscriptshiftGQ\"0F '-FB6/FfnF4FEF>FFFIFKFMFOFQFSFgnFin-F16'Q'𝕀F'F4F7F;F>F4F9FEF>-F[o 6%-F16'Q\"xF'F4F7F;F>-F#6$FcoFioFfo-FB6/Q\"=F'F4FEF>FFFIFKFMFOFQFS/FVQ ,0.2777778emF'/FYFip-I#mnGF$6&Q\"0F'F4FEF>Fio-FB6-Q\",F'/F " 0 "" {MPLTEXT 1 0 21 "d := Determinant(M);\n" }{MPLTEXT 1 0 26 "lambda := Eigenvalues(M);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "! \"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I'VectorG6$%*protectedGI(_sysli bG6\"6#I'columnGF(6#/I$%idGF(\")?j,a" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 74 "If also the eigenvectors are of interest, they can be obtained as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "lambda,ev : = Eigenvectors(M);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-&I'VectorG6$%*protectedGI(_syslibG6\"6#I'columnGF)6#/I$%idGF)\")Oe a_-I'MatrixGF&6#/F.\")'RzO&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 100 " To access these results individually, use \1673[i] and ev[1..-1,i] and not ev[i] (as one might expect); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "seq(\{lambda[i],ev[1..-1,i]\},i=1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%<$*$\"\"&#\"\"\"\"\"#-&I'VectorG6$%*protectedGI(_ syslibG6\"6#I'columnGF/6#/I$%idGF/\")OUz`<$,$F$!\"\"-F*6#/F4\")%y_P&<$ F'-F*6#/F4\")[Gv`" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 71 "We can chec k, that these results really satisfy the eigenvalue problem." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "MatrixVectorMultiply(M,ev[1. .-1,2])-lambda[2]*ev[1..-1,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I'Ve ctorG6$%*protectedGI(_syslibG6\"6#I'columnGF(6#/I$%idGF(\");/9b" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-&I'VectorG6$%*protectedGI(_syslibG6\"6#I'columnGF(6 #/I$%idGF(\")sXs^" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }