{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 24 " -----------------------" }}{PARA 0 "" 0 "" {TEXT -1 13 " c235-17.mws" }}{PARA 0 "" 0 "" {TEXT -1 23 "-----------------------" }}{PARA 0 "" 0 "" {TEXT -1 32 "Problem s on orthogonal subspaces" }}{PARA 0 "" 0 "" {TEXT -1 24 "and orthogon al matrices." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 " Q1:" }}{PARA 0 "" 0 "" {TEXT -1 45 "Find an o.n. basis fo r the space S spanned by" }}{PARA 0 "" 0 "" {TEXT -1 26 "the columns o f A, given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "A := <<1,3,3,1,3>|<1,2,1,0,1>|<0,1,2,1,2>>;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABLEG6%\")odp7-%'MATRIXG6#7 '7%\"\"\"F.\"\"!7%\"\"$\"\"#F.7%F1F.F27%F.F/F.F3%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 " We apply the QR decomposition and pick o ut the Q factor, calling it B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "B := QRDecomposition(A)[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"BG-%'RTABLEG6%\")%G)p7-%'MATRIXG6#7'7$,$*&\"#H!\"\"F0#\"\"\"\"\"#F3 ,$*(\"\")F3\"$$\\F1\"$')*F2F37$,$*(\"\"$F3F0F1F0F2F3,$*(\"#>F3F9F1F9F2 F37$F;,$*(\"\"&F3F8F1F9F2F17$F.,$*(\"#8F3F9F1F9F2F1FA%'MatrixG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 44 " The columns of B are the o.n. bas is sought." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 " Q2: By starting from the vector <1, 1, -1, 2, 2> construct a " }}{PARA 0 "" 0 "" {TEXT -1 55 " vector v that is orthogonal to \+ the space S of Q1." }}{PARA 0 "" 0 "" {TEXT -1 59 " [That is to say , v is orthogonal to every vector in S]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 " S2:" }}{PARA 0 "" 0 "" {TEXT -1 50 " We subtract from u that part of u that is in S." }} {PARA 0 "" 0 "" {TEXT -1 40 " To do this, we write v = u - B B^t u " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " u := <1, 1,-1,2,2>;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG-%'RTABLEG6%\")7iq7-%'MATRIXG6# 7'7#\"\"\"F-7#!\"\"7#\"\"#F1&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "v := u - B.Transpose(B).u;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'RTABLEG6%\")#H1F\"-%'MATRIXG6#7'7##\"#7\"#< 7##\"\"$\"#M7##!#LF07##\"#dF47##\"#=F0&%'VectorG6#%'columnG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 66 " We test to see if this is indeed orthogonal to the basis B for S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Transpose(B).v;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'RTABLEG6%\")sjq7-%'MATRIXG6#7$7#\"\"!F+&%'VectorG6#%'columnG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 62 " Thus we see that v is indeed ort hogonal to the basis B of S." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 " Q3: Find a basis for the orthogonal complement of S in R^5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 " S3: The projector P onto S is P = B B^t" }}{PARA 0 " " 0 "" {TEXT -1 65 " The projector Po onto the orthogonal spac e is therefore " }}{PARA 0 "" 0 "" {TEXT -1 62 " Po = I - P. The required o.n. basis Bo is given by" }}{PARA 0 "" 0 "" {TEXT -1 56 " the columns of the Q in the QR Decomposition of" }}{PARA 0 "" 0 "" {TEXT -1 61 " Po = I - B B^t. We need to use QR (of Gram-Schmidt)" }}{PARA 0 "" 0 "" {TEXT -1 49 " because the col umns of Po may not be o.n." }}{PARA 0 "" 0 "" {TEXT -1 53 " We \+ first find P and Po and show that they are" }}{PARA 0 "" 0 "" {TEXT -1 43 " projectors satisfying P + Po = I." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "P := B.Tra nspose(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'RTABLEG6%\")kdr 7-%'MATRIXG6#7'7'#\"\"&\"#<#\"\"(F0#!\"\"F0#!\"$F0F37'F1#\"#B\"#M#\"\" #F0#!\"&F:F;7'F3F;F1#\"\"%F0F17'F5F=F@#F2F:F@F?%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Po := IdentityMatrix(5) - P;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#PoG-%'RTABLEG6%\")g6s7-%'MATRIXG6#7 '7'#\"#7\"#<#!\"(F0#\"\"\"F0#\"\"$F0F37'F1#\"#6\"#M#!\"#F0#\"\"&F:F;7' F3F;#\"#5F0#!\"%F0F17'F5F=FB#\"#FF:FB7'F3F;F1FBF@%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "P.P,P;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")7Ps 7-%'MATRIXG6#7'7'#\"\"&\"#<#\"\"(F.#!\"\"F.#!\"$F.F17'F/#\"#B\"#M#\"\" #F.#!\"&F8F97'F1F9F/#\"\"%F.F/7'F3F;F>#F0F8F>F=%'MatrixG-F$6%\")kdr7F' FB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Po.Po,Po;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\");ls7-%'MATRIXG6#7'7'#\"#7\"#<#! \"(F.#\"\"\"F.#\"\"$F.F17'F/#\"#6\"#M#!\"#F.#\"\"&F8F97'F1F9#\"#5F.#! \"%F.F/7'F3F;F@#\"#FF8F@7'F1F9F/F@F>%'MatrixG-F$6%\")g6s7F'FF" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "P+Po;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")'**GF\"-%'MATRIXG6#7'7'\"\"\"\"\"!F-F-F- 7'F-F,F-F-F-7'F-F-F,F-F-7'F-F-F-F,F-7'F-F-F-F-F,%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "QRDecomposition(P)[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")/:t7-%'MATRIXG6#7'7$,$*&\"# " 0 "" {MPLTEXT 1 0 29 "Bo := QRDecomposition(Po)[1] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#BoG-%'RTABLEG6%\")7pt7-%'MATRI XG6#7'7%,$*(\"\"#\"\"\"\"#F3F?F5F3,$*&F0F3F0F 5F17%,$*&\"#MF3F4F5F1,$*&F0F3F?F5F1F67%FAFC,$*&F0F3F0F5F3%'MatrixG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Transpose(B).Bo;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")_$RF\"-%'MATRIXG6#7$7%\"\"!F, F,F+%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 " Thus we see tha t the columns of Bo are all orthogonal to the original basis B." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 " Q4: Find a diagonal matrix E whose diagonal en tries are the" }}{PARA 0 "" 0 "" {TEXT -1 24 "eigenvalues of M , where " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "M := <<4,-1,0>|<-1,4,0>|<0,0,3>>;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\")'*4+F-%'MATRIXG6#7%7%\"\"%!\"\" \"\"!7%F/F.F07%F0F0\"\"$%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 " S4: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalues(M); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\");:E8-%'MATRIXG6#7%7 #\"\"&7#\"\"$F-&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "E := DiagonalMatrix([5,3,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'RTABLEG6%\")#pOt#-%'MATRIXG6#7%7%\"\"&\"\"!F/7 %F/\"\"$F/7%F/F/F1%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 " Q 5: Find the largest and smallest values of the " }}{PARA 0 "" 0 "" {TEXT -1 46 " quadratic form q(x) = x^t M x, where " }}{PARA 0 "" 0 "" {TEXT -1 36 " x is a unit vector ||x|| = 1." }}{PARA 0 "" 0 "" {TEXT -1 4 " S5:" }}{PARA 0 "" 0 "" {TEXT -1 47 " The largest and smallest values of the " }}{PARA 0 "" 0 "" {TEXT -1 50 " \+ (constrained) quadratic form are the largest " }}{PARA 0 "" 0 "" {TEXT -1 59 " and smallest eigenvalues of M, that is to say, 5 and \+ 3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 " Q6: Find an orhtogonal matrix Q so that M = Q E Q^t and" }}{PARA 0 "" 0 "" {TEXT -1 38 " show that this is indeed the case." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " S6; We start with the \+ matrix V whose columns are the" }}{PARA 0 "" 0 "" {TEXT -1 52 " ei genvectors of M. Then we apply QR to T to " }}{PARA 0 "" 0 "" {TEXT -1 57 " generate the orthogonal matrix Q. This latter matrix " }}{PARA 0 "" 0 "" {TEXT -1 42 " is the one required, as we shall \+ show." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "V := Eigenvectors(M)[2 ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%'RTABLEG6%\")[e%)G-%'MAT RIXG6#7%7%!\"\"\"\"!\"\"\"7%F0F/F07%F/F0F/%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Q := QRDecomposition(V)[1];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"QG-%'RTABLEG6%\")kQKF-%'MATRIXG6#7%7%,$*&\" \"#!\"\"F0#\"\"\"F0F1\"\"!,$*&F0F1F0F2F37%F5F4F57%F4F3F4%'MatrixG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Q.E.Transpose(Q), M;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")K(y!H-%'MATRIXG6#7%7%\" \"%!\"\"\"\"!7%F-F,F.7%F.F.\"\"$%'MatrixG-F$6%\")'*4+FF'F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 " This confirms the representation M = Q E Q^t for M." }} {PARA 0 "" 0 "" {TEXT -1 97 "----------------------------------------- --------------------------------------------------------" }}}}{MARK "2 8 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 12695768 12698284 12706212 12706292 12706372 12715764 12721160 12723712 12726516 12728996 12731504 12736912 12739352 27000996 13261516 27336692 28845848 27323864 29078732 }{RTABLE M7R0 I5RTABLE_SAVE/12695768X,%)anythingG6"6"[gl!"%!!!#0"&"$"""""$F(F'F(F'""#F'""!F'F *F'F)F'F)6" } {RTABLE M7R0 I5RTABLE_SAVE/12698284X,%)anythingG6"6"[gl!"%!!!#+"&"#,$*$"#H#"""""##F+F),$F(#" "$F)F.F'F.,$*$"$')*F*#"")"$$\,$F2#"#>F3,$F2#!"&F6,$F2#!#8F3F:6" } {RTABLE M7R0 I5RTABLE_SAVE/12706212X*%)anythingG6"6"[gl!#%!!!"&"&"""F'!""""#F)6" } {RTABLE M7R0 I5RTABLE_SAVE/12706292X*%)anythingG6"6"[gl!#%!!!"&"&#"#7"#<#""$"#M#!#LF)#"#dF,# "#=F)6" } {RTABLE M7R0 I5RTABLE_SAVE/12706372X*%)anythingG6"6"[gl!#%!!!"#"#""!F'6" } {RTABLE M7R0 I5RTABLE_SAVE/12715764X,%)anythingG6"6"[gl!"%!!!#:"&"&#""&"#<#""(F)#!""F)#!"$F) F,F*#"#B"#M#""#F)#!"&F2F3F,F3F*#""%F)F*F.F5F7#F+F2F7F,F3F*F7F*6" } {RTABLE M7R0 I5RTABLE_SAVE/12721160X,%)anythingG6"6"[gl!"%!!!#:"&"&#"#7"#<#!"(F)#"""F)#""$F) F,F*#"#6"#M#!"#F)#""&F2F3F,F3#"#5F)#!"%F)F*F.F5F9#"#FF2F9F,F3F*F9F76" } {RTABLE M7R0 I5RTABLE_SAVE/12723712X,%)anythingG6"6"[gl!"%!!!#:"&"&#""&"#<#""(F)#!""F)#!"$F) F,F*#"#B"#M#""#F)#!"&F2F3F,F3F*#""%F)F*F.F5F7#F+F2F7F,F3F*F7F*6" } {RTABLE M7R0 I5RTABLE_SAVE/12726516X,%)anythingG6"6"[gl!"%!!!#:"&"&#"#7"#<#!"(F)#"""F)#""$F) F,F*#"#6"#M#!"#F)#""&F2F3F,F3#"#5F)#!"%F)F*F.F5F9#"#FF2F9F,F3F*F9F76" } {RTABLE M7R0 I5RTABLE_SAVE/12728996X,%)anythingG6"6"[gl!"%!!!#:"&"&"""""!F(F(F(F(F'F(F(F(F(F (F'F(F(F(F(F(F'F(F(F(F(F(F'6" } {RTABLE M7R0 I5RTABLE_SAVE/12731504X,%)anythingG6"6"[gl!"%!!!#+"&"#,$*$"#&)#"""""##F+"#<,$F( #""(F),$F(#!""F),$F(#!"$F)F2""!,$*$"#5F*#F+F;,$F:#F+""&F9F=6" } {RTABLE M7R0 I5RTABLE_SAVE/12736912X,%)anythingG6"6"[gl!"%!!!#0"&"$,$*$"#^#"""""##F,"#<,$F(# !"("$-",$F(#F+F2,$F(#F+"#MF3""!,$*$""$F*#F+""',$F:#!""F=,$F:F*F>F8F8,$*$F,F*F*F 8,$FC#F@F,6" } {RTABLE M7R0 I5RTABLE_SAVE/12739352X,%)anythingG6"6"[gl!"%!!!#'"#"$""!F'F'F'F'F'6" } {RTABLE M7R0 I5RTABLE_SAVE/27000996X,%)anythingG6"6"[gl!"%!!!#*"$"$""%!""""!F(F'F)F)F)""$F& } {RTABLE M7R0 I5RTABLE_SAVE/13261516X*%)anythingG6"6"[gl!#%!!!"$"$""&""$F(F& } {RTABLE M7R0 I5RTABLE_SAVE/27336692X,%)anythingG6"6"[gl!"%!!!#*"$"$""&""!F(F(""$F(F(F(F)F& } {RTABLE M7R0 I5RTABLE_SAVE/28845848X,%*algebraicG6"6"[gl!"%!!!#*"$"$!"""""""!F)F)F(F(F(F)F& } {RTABLE M7R0 I5RTABLE_SAVE/27323864X,%)anythingG6"6"[gl!"%!!!#*"$"$,$*$""##"""F)#!""F),$F(F* ""!F/F/F+F.F.F/F& } {RTABLE M7R0 I5RTABLE_SAVE/29078732X,%)anythingG6"6"[gl!"%!!!#*"$"$""%!""""!F(F'F)F)F)""$F& }