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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "L-11 MCS 563 Friday 17 Sep
tember 2004" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
173 "This Maple worksheet illustrates the calculation of the Jordan ca
nonical form for the polynomial of Example 5.5, which is further discu
ssed in section 5.1.4 of the textbook." }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p := (x-1)^3*x*(x+1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"pG*(),&%\"xG\"\"\"F)!\"\"\"\"$F)F(F),&F(F)F)
F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "This is an exact (or int
rinsic) polynomial, so the answers Maple will give us will be exact as
 well." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The \+
multipliciation matrix is known as the companion matrix:" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "C := LinearAlgebra
[CompanionMatrix](p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'RTAB
LEG6$\"*s%[`8-%'MATRIXG6#7'7'\"\"!F.F.F.F.7'\"\"\"F.F.F.F07'F.F0F.F.!
\"#7'F.F.F0F.F.7'F.F.F.F0\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 
"The roots of the polynomial are eigenvalues of the companion matrix:
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "z :=
 LinearAlgebra[Eigenvalues](C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"zG-%'RTABLEG6$\"*WK!y8-%'MATRIXG6#7'7#\"\"!7#!\"\"7#\"\"\"F1F1" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "Because we have a triple root, th
e companion matrix cannot be diagonalized.  The closest form to a diag
onal form is the so-called Jordan canonical form:" }{MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "J := LinearAlgebra[Jordan
Form](C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'RTABLEG6$\"*?X<N
\"-%'MATRIXG6#7'7'\"\"!F.F.F.F.7'F.!\"\"F.F.F.7'F.F.\"\"\"F2F.7'F.F.F.
F2F27'F.F.F.F.F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 266 "Observe that
 on the diagonal we find the roots of the polynomial (or the eigenvalu
es of the companion matrix).  The triple root corresponds to a 3-by-3 \+
Jordan block.  While J reveals the eigenvalues, the eigenvectors (or g
eneralizations thereof) are in the matrix Q:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 45 "Q := LinearAlgebra[JordanForm](C,output='Q');" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG-%'RTABLEG6$\"*?]6N\"-%'MATRIXG
6#7'7'\"\"\"\"\"!F/F/F/7'!\"##!\"\"\"\")#F.\"\"##!\"&\"\"%#\"#<F47'F/#
\"\"$F4#F3F6#F>F9#!\"$F47'F6FAF?#\"\"&F9#!#8F47'F3#F.F4F5#FBF9#\"\"(F4
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "Since the first two roots ar
e regular, we have the normal eigenvectors.  The first column is a vec
tor in the kernel of C, corresponding to the first eigenvalue zero:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "v1 := V
ector(LinearAlgebra[Column](Q,1)): C.v1 = 0*v1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'RTABLEG6$\"*+0\"[8-%'MATRIXG6#7'7#\"\"!F,F,F,F,F-" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "The second column of Q gives th
e eigenvector corresponding to the eigenvalue -1:" }{MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "v2 := Vector(LinearAlgebr
a[Column](Q,2)): C.v2 = (-1)*v2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-
%'RTABLEG6$\"*CS*y8-%'MATRIXG6#7'7#\"\"!7##\"\"\"\"\")7##!\"$F17##\"\"
$F17##!\"\"F1-F%6$\"*W_.N\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "
Below we see the Q matrix as bringing C to its normal form:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Q^(-1).
C.Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*KY1N\"-%'MATRIX
G6#7'7'\"\"!F,F,F,F,7'F,!\"\"F,F,F,7'F,F,\"\"\"F0F,7'F,F,F,F0F07'F,F,F
,F,F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Since J is not diagonal
, the three last columns of Q are not eigenvectors in the usual sense,
 they are generalized eigenvectors." }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 90 "The following relation is the eigenvalue-
eigenvector equation C*x=lambda*x in matrix form:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 10 "C.Q = Q.J;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%'RTABLEG6$\"*;[uP\"-%'MATRIXG6#7'7'\"\"!F-F-F-F-7'F-#\"\"\"\"\")#
F0\"\"##!\"$\"\"%#\"\"(F17'F-#F5F1#!\"\"F3#F0F6#\"\"$F17'F-F>F;#F?F6F:
7'F-#F<F1F2#F<F6F/-F%6$\"*#>n[8F(" }}}{EXCHG {PARA 0 "> " 0 "" 
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