{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 " ------------------------- --------" }}{PARA 0 "" 0 "" {TEXT -1 9 " c235-15" }}{PARA 0 "" 0 "" {TEXT -1 33 "---------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 49 "Orthogonal transformations on a real vector space" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Defn. T \+ is orthogonal if = for all x, y in the space" }} {PARA 0 "" 0 "" {TEXT -1 44 "That is to say, T preserves inner produc ts." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "De fn. Adjoint T* of T = " }}{PARA 0 "" 0 "" {TEXT -1 40 "Theorem: in an o.n. basis [T*] = [T]^t" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Defn. T is self adjoi nt if T* = T <==> [T] = [T]^t" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 40 "Summary (in matrix terms wrt o.n. bases) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "(1) \+ (Ax)^t (Ay) = x^t y, for all x,y, iff A^t A = I (ie A is orthog onal)" }}{PARA 0 "" 0 "" {TEXT -1 60 "(2) (Ax)^t y = x^t (A^t)y. We say A^t is the adjoint of A" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " We create some test matrices" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(Line arAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A := <<1,2,3 ,0>|<0,2,1,3>|<1,1,0,1>|<0,1,0,1>>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"AG-%'RTABLEG6%\")'*Qp7-%'MATRIXG6#7&7&\"\"\"\"\"!F.F/7&\"\"#F1F.F .7&\"\"$F.F/F/7&F/F3F.F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "B := <<1,2,3,0>|<2,3,-1,0>|<3,-1,2,1>|<0,0,1,4>>;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6%\");kp7-%'MATRIXG6#7 &7&\"\"\"\"\"#\"\"$\"\"!7&F/F0!\"\"F17&F0F3F/F.7&F1F1F.\"\"%%'MatrixG " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 " B is symmetric B^t = B" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "C := <<1/2,1/2,1/2,1/2>|<1/sqrt(2), -1/sqrt(2),0,0>|\n <0, 0,1/sqrt(2), -1/sqrt(2)>|<1/2,1/2,-1/2,-1/2>>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'RTABLEG6%\");]q7-%'MATRIXG6#7&7&#\"\"\"\"\"#,$ *&F0!\"\"F0F.F/\"\"!F.7&F.,$*&F0F3F0F.F3F4F.7&F.F4F1#F3F07&F.F4F6F9%'M atrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 " C is orthogonal" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Transpose(C).C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")!34F\"-%'MATRIXG6#7&7&\"\"\"\"\" !F-F-7&F-F,F-F-7&F-F-F,F-7&F-F-F-F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x := <1,2,3,4>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'RTABLEG6%\")wmp7-%'MATRIXG6#7&7#\"\"\"7#\"\"#7#\"\"$7#\"\" %&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y := <0,2,3,-1>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-%'RTABLEG6% \")cnp7-%'MATRIXG6#7&7#\"\"!7#\"\"#7#\"\"$7#!\"\"&%'VectorG6#%'columnG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ip:=(u,v)->simplify(Tra nspose(u).v):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "At := Tran spose(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#AtG-%'RTABLEG6%\")gcr7 -%'MATRIXG6#7&7&\"\"\"\"\"#\"\"$\"\"!7&F1F/F.F07&F.F.F1F.7&F1F.F1F.%'M atrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ip(A.x,y), ip(x, \+ At.y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#GF#" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 29 " The adjoint of A is A^t = At" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ip(B.x,y), \+ ip(x,B.y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#CF#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 " B = B^t , ie B is self adjoint" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ip(x,y), ip(C.x, C.y); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"*F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 " Th e orthogonal transformation (represented here by an orthogonal matrix) " }}{PARA 0 "" 0 "" {TEXT -1 29 " preserves the inner product" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 " \+ We design a Euclidean distance function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "d:=(u,v)->sqrt(ip(u-v, u-v)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "d(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*$\"#E#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "d(x,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "d(y,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"#E# \"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "d(A.x, A.y), d(x,y);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$,$*&\"\"#\"\"\"\"#@#F&F%F&*$\"#EF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "d(B.x, B.y), d(x,y);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$*$\"$p%#\"\"\"\"\"#*$\"#EF%" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "d(C.x,C.y), d(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*$\"#E#\"\"\"\"\"#F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 " Distance is preserved by orhtogonal transformations such as C " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Rotat ions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "R:=(t)-><|<-sin(t), cos(t)>>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RGj+6#% \"tG6\"6$%)operatorG%&arrowGF(-%$<|gr>G6$-%$<,>G6$-%$cosG6#9$-%$sinGF4 -F06$,$F6!\"\"F2F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "R(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")+%>F\"-%' MATRIXG6#7$7$-%$cosG6#%\"tG,$-%$sinGF.!\"\"7$F1F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "u := <1,2>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'RTABLEG6%\");pp7-%'MATRIXG6#7$7#\"\"\"7#\"\"#& %'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "v : = <4,-1>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'RTABLEG6%\")'*pp 7-%'MATRIXG6#7$7#\"\"%7#!\"\"&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ip(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ip(R(t).u, R(t).v) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "ip(R(0.3).u, R(0.3).v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++?!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "R(t).u;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")wqp7-%' MATRIXG6#7$7#,&-%$cosG6#%\"tG\"\"\"*&\"\"#F1-%$sinGF/F1!\"\"7#,&F4F1*& F3F1F-F1F1&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "R(t).v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")crp7- %'MATRIXG6#7$7#,&*&\"\"%\"\"\"-%$cosG6#%\"tGF/F/-%$sinGF2F/7#,&*&F.F/F 4F/F/F0!\"\"&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify(Determinant(R(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "si mplify(Determinant(C));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 " Actually, orhogonal matrices hav e det =1 or -1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Reflection" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "F := <<0,1>|<1,0>>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'RTABLEG6%\")#RNF\"-%'MATRIXG6#7$7$\"\"!\" \"\"7$F/F.%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "u, F. u;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\");pp7-%'MATRIXG6#7 $7#\"\"\"7#\"\"#&%'VectorG6#%'columnG-F$6%\")wsp7-F(6#7$F-F+F/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "v, F.v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")'*pp7-%'MATRIXG6#7$7#\"\"%7#!\"\"&%'Vect orG6#%'columnG-F$6%\")ctp7-F(6#7$F-F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Transpose(F).F;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")WVu7-%'MATRIXG6#7$7$\" \"\"\"\"!7$F-F,%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 " F i s an orthogonal matrix with determinant -1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 " We can rotate and reflect (a \+ product of orthogonal matrices is orthogonal)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ip(R(t).F.u, R(t).F.v), ip(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "d(R(t).u, R(t).v), d(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*& \"\"$\"\"\"\"\"##F&F'F&F#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 " Dis tance is preserved by an isomorphism, that is a shift w and an" }} {PARA 0 "" 0 "" {TEXT -1 26 " orthogonal transformation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 11 "w := <3,5>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG-%'RTABLEG6%\")wup7-%'MATRIXG6#7$7#\"\"$7#\"\"&&%'VectorG6#%' columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "d(w + R(t).F.u, \+ w + R(t).F.v), d(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&\"\"$\" \"\"\"\"##F&F'F&F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "d(w - R(0.7).u, w - R(0.7).v);" }}{PARA 11 "" 1 "" {XPPMATH 20 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