{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "Helvetica" 1 14 128 0 0 1 0 0 2 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 24 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "He lvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Headin g 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 " " 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 } {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 18 256 1 {CSTYLE "" -1 -1 "" 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 257 55 "Eigenvalues, Eigenvectors and Quadratic Forms: Examples " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 19 "Mth 254 Mar 12 2001" }}{PARA 0 "" 0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 259 39 "Filename: 254w2001_eigenvects_examp.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 48 "Linear Algebra supplem ent for Vector Calculus 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 233 "In this worksheet I give a few examples of eigen values and eigenvectors of matrices in the first part. In the second p art I provide the answeres, as given by Maple, to the problems in the \+ Mth 254 Study Guide (Parks 2000-01 revision)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "If you try a few of thes e examples by hand you will develop a deep appreciation for Maple, or \+ a similar tool. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning , new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 16 "Part 1. \+ Examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A1:=matrix(3,3,[-1,2,2,2,2,2,-3,-6,-6]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'matrixG6#7%7%!\"\"\"\"#F+7%F+ F+F+7%!\"$!\"'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenva ls(A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!!\"$!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "A1 has distinct eigenvalues. Thus A1 is d iagonizable. Let's diagonalize it explicitly." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "v1:=eigenvects(A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v1G6%7%!\"$\"\"\"<#-%'vectorG6#7%!\"\"\"\"!F(7%F/F(< #-F+6#7%F/F.F(7%!\"#F(<#-F+6#7%F6F(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "s1:=\{\}: for k from 1 to 3 do s1:=s1 union op(3,op(k ,[v1])): od: unassign('k'): s1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%- %'vectorG6#7%!\"\"\"\"!\"\"\"-F%6#7%F)F(F*-F%6#7%!\"#F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "S1:=matrix(3,0,[]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S1G-%&arrayG6%;\"\"\"\"\"$;F)\"\"!7\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "S1:=augment(S1,op(s1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S1G-%'matrixG6#7%7%\"\"!!\"\"!\"#7% F+F*\"\"\"7%F.F.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "inver se(S1) &* A1 &* S1: D1:=evalm(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#D1G-%'matrixG6#7%7%\"\"!F*F*7%F*!\"$F*7%F*F*!\"#" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 35 "Alternate Calculation for Example 1" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "There is an easie r way to do these calculations. Let's start over. First we compute the eigenvaules of A1 and put them in a diagonal matrix. We can do this a s follows:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "D1b:=diag(eigenvals(A1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$D1bG-%'matrixG6#7%7%\"\"!F*F*7%F*!\"$F*7%F*F*!\"#" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 " Now if A1 is diagonizable the it will be 'similar' to D1b - so we ask Maple to check:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "issimilar(A1,D1b,'P1b');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 161 "The return value indicates A1 is diago nizable. In this case P1b has been assigned a matrix which plays the role of the inverse of S1 above. let's check it;" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "evalm(P1b ); S1b:=inverse(P1b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7 %7%#\"\"\"\"\"'#F)\"\"$F(7%#!\"\"F,#!\"#F,F07%F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$S1bG-%'matrixG6#7%7%\"\"!\"\"$!\"#7%\"\"'F*\"\"\" 7%!\"'!\"$F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 56 "It doesn't look the same, but let's check that it works :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "inverse(S1b) &* A1 &* S1b: evalm(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!F(F(7%F(!\"$F(7%F(F(!\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "That worked as expected." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "A2:=matrix(3,3,[7,4,-4,4,-8,-1,-4,-1,-8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G-%'matrixG6#7%7%\"\"(\"\"%!\"%7%F+!\")!\"\"7%F,F/ F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvals(A2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"*!\"*F$" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "This time we have an e igenvalue (-9) with algebraic multiplicity 2. We do not yet know if A2 is diagonizable or not. Let's compute the eigenvectors." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "v2:= eigenvects(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v2G6$7%\"\"*\"\" \"<#-%'vectorG6#7%\"\"%F(!\"\"7%!\"*\"\"#<$-F+6#7%\"\"!F(F(-F+6#7%F(! \"%F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "We see that -9 has geometric multiplicity 2. Thus A is d iagonizable. Let's diagonalize it explicitly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "s2:=op(3,op (1,[v2])) union op(3,op(2,[v2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#s2G<%-%'vectorG6#7%\"\"!\"\"\"F+-F'6#7%F+!\"%F*-F'6#7%\"\"%F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "S2:=matrix(3,0,[]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S2G-%&arrayG6%;\"\"\"\"\"$;F)\"\"!7 \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "S2:=augment(S2,op(s2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S2G-%'matrixG6#7%7%\"\"%\"\" \"\"\"!7%F+!\"%F+7%!\"\"F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "inverse(S2) &* A2 &* S2: D2:=evalm(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#D2G-%'matrixG6#7%7%\"\"*\"\"!F+7%F+!\"*F+7%F+F+F-" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Again, as expected." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A 3:=matrix(3,3,[1,1,-1,-1,3,-1,-1,2,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G-%'matrixG6#7%7%\"\"\"F*!\"\"7%F+\"\"$F+7%F+\"\"#\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eigenvals(A3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#\"\"\"F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "Again we have an eigenvalue (1 ) with algebraic multiplicity 2. We do not yet know if A3 is diagoniza ble or not. Let's compute the eigenvectors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s3:=eigenvects(A3 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s3G6$7%\"\"\"\"\"#<#-%'vector G6#7%F'F'F'7%F(F'<#-F+6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "Oops. We see the eigenvalue 1 \+ has geometric multiplicity 1 (only one eigenvector for 1 is listed by \+ Maple). Thus A3 is not diagonizable. The \"closest\" we can get it the Jordan canonical form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "jordan(A3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"#\"\"!F)7%F)\"\"\"F+7%F)F)F+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 168 "Y ou probably suspect we could have used the jordan() function above to \+ verify the diagonizability of A1 and A2. If so, you are correct, but i t would have been less fun." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "jordan(A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%!\"#\"\"!F)7%F)!\"$F)7%F)F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "jordan(A2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%!\"*\"\"!F)7%F)\"\"*F)7%F)F)F(" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 9 "Ex ample 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "This time consider a symmetric matrix. We know that a symmetric m atrix has real eigenvalues and is diagonizable whether it has repeated eigenvalues or not." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "A4:=matrix(4,4,[0,1,-1,2,1,1,0,-1,-1,0,-1 ,1,2,-1,1,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A4G-%'matrixG6#7& 7&\"\"!\"\"\"!\"\"\"\"#7&F+F+F*F,7&F,F*F,F+7&F-F,F+F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "If we co mpute the exact eigenvalues we get a nearly incomprehensible expressio n, so let's compute approximate eigenvalues instead." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(eig envals(A4),16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"\"#\"\"!$\"1x\\g V1L\"o\"!#:,&$!1yk3wUSBLF(\"\"\"%\"IG$!\"#!#;,&$!0+&=vOEzNF(F,F-F." }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 " Those imaginary parts should not be there! They are due to roundoff as you can see by doing the calculation at several precisions. For examp le" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(eigenvals(A4),22);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&$\"\"#\"\"!,&$\"7vaPx\\gV1L\"o\"!#@\"\"\"%\"IG$!\"\"!#A ,&$!7sdixk3wUSBLF)F*F+F,,&$!6/(\\(*\\=vOEzNF)F*F+F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "That A4 is diag onizable we can verify by computing the Jordan canonical form" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(jordan(A4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG 6#7&7&$\"\"#\"\"!F*F*F*7&F*,&$\"+V1L\"o\"!\"*\"\"\"%\"IG$!\"$!#5F*F*7& F*F*$!+xUSBLF/F*7&F*F*F*$!*nj#zNF/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Here is another symmetric example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 50 "A5:=matrix(4,4,[0,1,2,3,1,0,1,2,2,1,0,1,3,2,1,0]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A5G-%'matrixG6#7&7&\"\"!\"\"\" \"\"#\"\"$7&F+F*F+F,7&F,F+F*F+7&F-F,F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "jordan(A5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'mat rixG6#7&7&,&!\"#\"\"\"*$-%%sqrtG6#\"\"#\"\"\"F*\"\"!F1F17&F1,&F)F*F+! \"\"F1F17&F1F1,&F/F**$-F-6#\"#5F0F*F17&F1F1F1,&F/F*F7F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "We see A 5 is diagonizable as expected." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT -1 48 "Part 2. Solutions for the Study Guide, Le sson 22" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 417 "Here are Maple's solutions for the problems in Lesson 22. The mat rices are numbered in accord with the problem numbers. You may note so me trivial differences between the solutions and the selected solution s in the Study Guide. Keep in mind any nonzero multiple of an eigenvec tor is an eigenvector. More generally, any nonzero linear combination \+ of eigenvectors corresponding to the same eigenvalue, is an eigenvecto r." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 174 "Af ter studying the examples above you should be able to decipher Maple's output. Explicitly the eigenvects() function returns a list consistin g of ordered triples of the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 80 "[eigenvalue, algebraic multiplicity, \{ linearly independent set of eigenvectors\}]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 222 "The number of vectors in the r eturned set of eigenvectors is the geometric multiplicity of the eigen value. The matrix is diagonizable if and only if the algebraic and geo metric multiplicities are equal for each eigenvalue." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "M01:=matr ix(2,2,[6,2,1,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M01G-%'matrix G6#7$7$\"\"'\"\"#7$\"\"\"\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvects(M01);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"%\" \"\"<#-%'vectorG6#7$!\"\"F%7%\"\"(F%<#-F(6#7$\"\"#F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "M02:=matrix(2,2,[-4,1,2,-3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M02G-%'matrixG6#7$7$!\"%\"\"\"7$\"\"#!\"$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvects(M02);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$7%!\"&\"\"\"<#-%'vectorG6#7$!\"\"F%7%! \"#F%<#-F(6#7$F%\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "M0 3:=matrix(2,2,[4,-1,5,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M03G- %'matrixG6#7$7$\"\"%!\"\"7$\"\"&\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvects(M03);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7 %,&\"\"$\"\"\"%\"IG\"\"#F&<#-%'vectorG6#7$,&#F&\"\"&F&F'#F(F0F&7%,&F%F &F'!\"#F&<#-F+6#7$,&F/F&F'#F4F0F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "M04:=matrix(2,2,[-5,1,-5,-3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M04G-%'matrixG6#7$7$!\"&\"\"\"7$F*!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvects(M04);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$7%,&!\"%\"\"\"%\"IG\"\"#F&<#-%'vectorG6#7$F&,&F&F&F'F (7%,&F%F&F'!\"#F&<#-F+6#7$F&,&F&F&F'F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "M05:=matrix(2,2,[-6,2,2,-3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M05G-%'matrixG6#7$7$!\"'\"\"#7$F+!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvects(M05);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$7%!\"#\"\"\"<#-%'vectorG6#7$F%\"\"#7%!\"(F%<#-F(6#7$F $F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "M06:=matrix(2,2,[6,2 ,2,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M06G-%'matrixG6#7$7$\"\" '\"\"#7$F+\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvec ts(M06);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7%\"\"#\"\"\"<#-%'vectorG6 #7$F%!\"#7%\"\"(F%<#-F(6#7$F$F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "M07:=matrix(3,3,[5,2,2,2,-2,0,2,0,-2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M07G-%'matrixG6#7%7%\"\"&\"\"#F+7%F+!\"#\"\"!7%F+F.F -" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvects(M07);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%7%!\"$\"\"\"<#-%'vectorG6#7%F%!\"#F+7% F+F%<#-F(6#7%\"\"!!\"\"F%7%\"\"'F%<#-F(6#7%\"\"%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "M08:=matrix(3,3,[5,3,3,3,2,0,3,0,2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M08G-%'matrixG6#7%7%\"\"&\"\"$F+7%F +\"\"#\"\"!7%F+F.F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eige nvects(M08);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%\"\"#\"\"\"<#-%'vect orG6#7%\"\"!!\"\"F%7%\"\")F%<#-F(6#7%F$F%F%7%F,F%<#-F(6#7%F%F,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "M09:=matrix(3,3,[-7,0,3,0,-7 ,2,3,2,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M09G-%'matrixG6#7%7% !\"(\"\"!\"\"$7%F+F*\"\"#7%F,F.\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvects(M09);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7 %\"\"'\"\"\"<#-%'vectorG6#7%#\"\"$\"\"#F%#\"#8F-7%!\")F%<#-F(6#7%!\"$! \"#F%7%!\"(F%<#-F(6#7%F%#F6F-\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "M10:=matrix(3,3,[1,0,1,0,1,2,1,2,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M10G-%'matrixG6#7%7%\"\"\"\"\"!F*7%F+F*\"\"#7%F *F-\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eigenvects(M10) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7%\"\"'\"\"\"<#-%'vectorG6#7%F%\" \"#\"\"&7%\"\"!F%<#-F(6#7%!\"\"!\"#F%7%F%F%<#-F(6#7%F4F%F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 15 "Quadratic Forms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 427 "Once we know the eigenvalues of the symmetric matrix of a quadrat ic form we can write down the diagonalized version of the quadratic fo rm. In general we are after more. We want to know an explicit change o f variables that diagonalizes the quadratic form. With a little coachi ng, Maple can solve this sort of problem. I'll show you one way of doi ng it here, but there may be better ways. If you use Maple, you should experiment." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Q11:=-6*x^2+4*x*y-3*y^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Q11G,(*$)%\"xG\"\"#\"\"\"!\"'*&F(\"\"\"%\"yGF-\"\"%* $)F.F)F*!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "M11:=hessia n(Q11/2,[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M11G-%'matrixG6# 7$7$!\"'\"\"#7$F+!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "D1 1:=diag(eigenvals(M11));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$D11G-%' matrixG6#7$7$!\"(\"\"!7$F+!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "issimilar(M11,D11,P11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%t rueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "We would like to use the inverse of P11 to define our change \+ of variables, but we have to normalize the columns of the inverse firs t. Let's call the result S11" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "normalize(col(inverse(P11),1 )),normalize(col(inverse(P11),2)): S11:=augment(matrix(2,0,[]),%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$S11G-%'matrixG6#7$7$,$*$-%%sqrtG6# \"\"&\"\"\"#!\"#F/,$F+#\"\"\"F/7$F3,$F+#\"\"#F/" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "sub11:=evalm(S11 &* matrix(2,1,[u,v]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sub11G-%'matrixG6#7$7#,&*&-%%sqrtG6 #\"\"&\"\"\"%\"uG\"\"\"#!\"#F/*&F,F0%\"vGF2#F2F/7#,&F+F7F5#\"\"#F/" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "subs(x=sub11[1,1], y=sub11[ 2,1], Q11); 'Q11'=expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$), &*&-%%sqrtG6#\"\"&\"\"\"%\"uG\"\"\"#!\"#F+*&F(F,%\"vGF.#F.F+\"\"#F,!\" '*&F&F.,&F'F3F1#F4F+F.\"\"%*$)F7F4F,!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$Q11G,&*$)%\"uG\"\"#\"\"\"!\"(*$)%\"vGF)F*!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "We have s uccessfully diagonalized the quadratic form Q11. Let's try another one ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Q12:=6*x^2+4*x*y+3*y^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Q12G,(*$)%\"xG\"\"#\"\"\"\"\"'*&F(\"\"\"%\"yGF-\"\"%*$)F.F)F* \"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "M12:=hessian(Q12/2 ,[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$M12G-%'matrixG6#7$7$\" \"'\"\"#7$F+\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "D12:=d iag(eigenvals(M12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$D12G-%'matr ixG6#7$7$\"\"#\"\"!7$F+\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "issimilar(M12,D12,P12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tru eG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 247 "We already knew t hat M12 and D12 are similar (because M12 is symmetric and so dia gonizable). The only reason for asking Maple about it is that is a con venient way to assign the desired value to P12 (just as above - see al so Example 1 above)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "normalize(col(inverse(P12),1)),normalize( col(inverse(P12),2)): S12:=augment(matrix(2,0,[]),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$S12G-%'matrixG6#7$7$,$*$-%%sqrtG6#\"\"&\"\"\"#! \"\"F/,$F+#\"\"#F/7$F3,$F+#\"\"\"F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sub12:=evalm(S12 &* matrix(2,1,[u,v]));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&sub12G-%'matrixG6#7$7#,&*&-%%sqrtG6#\"\"&\"\" \"%\"uG\"\"\"#!\"\"F/*&F,F0%\"vGF2#\"\"#F/7#,&F+F7F5#F2F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "subs(x=sub12[1,1], y=sub12[2,1], Q1 2); 'Q12'=expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$),&*&-%%sq rtG6#\"\"&\"\"\"%\"uG\"\"\"#!\"\"F+*&F(F,%\"vGF.#\"\"#F+F4F,\"\"'*&F&F .,&F'F3F1#F.F+F.\"\"%*$)F7F4F,\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%$Q12G,&*$)%\"uG\"\"#\"\"\"F)*$)%\"vGF)F*\"\"(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "That finishes the problems in the study guide, but let's also consider a quadratic form in 3 variables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Q:=3*x^2+12*x*y+24*y*z-6*x*z+6*y^2+6*z^2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG,.*$)%\"xG\"\"#\"\"\"\"\"$*&F(\" \"\"%\"yGF-\"#7*&F.F*%\"zGF-\"#C*&F(F*F1F*!\"'*$)F.F)F*\"\"'*$)F1F)F*F 7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "M:=hessian(Q/2,[x,y,z] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7%7%\"\"$\"\"' !\"$7%F+F+\"#77%F,F.F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "D 0:=diag(eigenvals(M));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#D0G-%'mat rixG6#7%7%\"\"'\"\"!F+7%F+,&#\"\"*\"\"#\"\"\"*$-%%sqrtG6#\"#&)\"\"\"# \"\"$F0F+7%F+F+,&F.F1F2#!\"$F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "issimilar(M,D0,P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "normalize(col(inverse( P),1)),normalize(col(inverse(P),2)),normalize(col(inverse(P),3)): S:=a ugment(matrix(3,0,[]),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG-%' matrixG6#7%7%,$*$-%%sqrtG6#\"#@\"\"\"#\"\"%F/,$*&F0F0*$-F-6#,&\"$])\" \"\"*$-F-6#\"#&)F0\"#uF0!\"\"!\"',$*&F0F0*$-F-6#,&F9F:F;!#uF0F@\"\"'7% ,$F+#F:F/,$*&*&,&\"# " 0 "" {MPLTEXT 1 0 37 "sub:=evalm(S &* matrix(3,1, [u,v,w]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$subG-%'matrixG6#7%7#, (*&-%%sqrtG6#\"#@\"\"\"%\"uG\"\"\"#\"\"%F/*&%\"vGF0*$-F-6#,&\"$])F2*$- F-6#\"#&)F0\"#uF0!\"\"!\"'*&%\"wGF0*$-F-6#,&F;F2F " 0 "" {MPLTEXT 1 0 48 "T:=subs(x=sub[1,1], y=sub [2,1], z=sub[3,1], Q); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,.*$) ,(*&-%%sqrtG6#\"#@\"\"\"%\"uG\"\"\"#\"\"%F-*&%\"vGF.*$-F+6#,&\"$])F0*$ -F+6#\"#&)F.\"#uF.!\"\"!\"'*&%\"wGF.*$-F+6#,&F9F0F:!#uF.F?\"\"'\"\"#F. \"\"$*&F(F0,(F)#F0F-*&*(,&\"# " 0 "" {MPLTEXT 1 0 13 "T:=expand(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,,*$)%\"u G\"\"#\"\"\"\"\"'*&*&)%\"vGF)F*-%%sqrtG6#\"#&)F*F*,&\"$])\"\"\"*$F0F* \"#u!\"\"\"%3;*&*&)%\"wGF)F*F0F*F*,&F5F6F7!#uF9!%3;*&*$F.F*F*F4F9\"&gK \"*&*$F=F*F*F?F9FD" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Now it is obvious, but it would be nice if we cou ld collect the terms as well:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "collect(T,[u,v,w]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"uG\"\"#\"\"\"\"\"'*&,&*&*$-%%s qrtG6#\"#&)F(F(,&\"$])\"\"\"*$F.F(\"#u!\"\"\"%3;*&F(F(F2F7\"&gK\"F4)% \"vGF'F(F4*&,&*&F(F(,&F3F4F5!#uF7F:*&*$F.F(F(F@F7!%3;F4)%\"wGF'F(F4" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 " The coefficients do not look like the original eigenvalues, so let's c heck them:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "c2:=subs(u=0,v=1,w=0,T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G,&*&*$-%%sqrtG6#\"#&)\"\"\"F,,&\"$])\"\"\"*$F(F, \"#u!\"\"\"%3;*&F,F,F-F2\"&gK\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "is(c2=D0[2,2]);;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "c3:=subs(u=0,v=0,w=1,T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c3G,&*&\"\"\"F',&\"$])\"\"\"*$-%% sqrtG6#\"#&)F'!#u!\"\"\"&gK\"*&*$F,F'F'F(F1!%3;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "is(c3=D0[3,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "In spite of appearances the coefficients are correct. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "0 1 0" 45 }{VIEWOPTS 1 1 0 1 1 1803 }