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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "    Find the diago
nalization of the matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A:=matrix(3,3,[0,0,.3,.4,0,0,0,.7,.
9]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%\"\"!F*$
\"\"$!\"\"7%$\"\"%F-F*F*7%F*$\"\"(F-$\"\"*F-" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Wa
rning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning
, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "u:=eigenvects(A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"uG6%7%$\"
+.OUj)*!#5\"\"\"<#-%'vectorG6#7%$\"+%4B-E$F)$\"+Qm9A8F)$\"+4()*=2\"!\"
*7%,&$!+S!=rJ%!#6F*%\"IG$\"+TJ;')GF)F*<#-F-6#7%,&$!+JXhkQF)F*F<$!+G-h$
3)F),&$!+veW<5F6F*F<$\"+$>_z(oF),&$\"+ee+L$)F)F*F<$!+QgqaDF)7%,&F9F*F<
$!+TJ;')GF)F*<#-F-6#7%,&FDF*F<$\"+G-h$3)F),&FIF*F<$!+$>_z(oF),&FNF*F<$
\"+QgqaDF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "u[1];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7%$\"+.OUj)*!#5\"\"\"<#-%'vectorG6#7%$
\"+%4B-E$F&$\"+Qm9A8F&$\"+4()*=2\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "u[2];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%,&$!+S!=rJ%
!#6\"\"\"%\"IG$\"+TJ;')G!#5F(<#-%'vectorG6#7%,&$!+JXhkQF,F(F)$!+G-h$3)
F,,&$!+veW<5!\"*F(F)$\"+$>_z(oF,,&$\"+ee+L$)F,F(F)$!+QgqaDF," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "u[3];" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7%,&$!+S!=rJ%!#6\"\"\"%\"IG$!+TJ;')G!#5F(<#-%'vectorG6#
7%,&$!+JXhkQF,F(F)$\"+G-h$3)F,,&$!+veW<5!\"*F(F)$!+$>_z(oF,,&$\"+ee+L$
)F,F(F)$\"+QgqaDF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x:=ar
ray(1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%&arrayG6$;\"\"\"
\"\"$7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "x[1]:=u[1][3][1
];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"-%'vectorG6#7%$\"
+%4B-E$!#5$\"+Qm9A8F.$\"+4()*=2\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "x[2]:=u[2][3][1];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
>&%\"xG6#\"\"#-%'vectorG6#7%,&$!+JXhkQ!#5\"\"\"%\"IG$!+G-h$3)F/,&$!+ve
W<5!\"*F0F1$\"+$>_z(oF/,&$\"+ee+L$)F/F0F1$!+QgqaDF/" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "x[3]:=u[3][3][1];" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>&%\"xG6#\"\"$-%'vectorG6#7%,&$!+JXhkQ!#5\"\"\"%\"IG$\"
+G-h$3)F/,&$!+veW<5!\"*F0F1$!+$>_z(oF/,&$\"+ee+L$)F/F0F1$\"+QgqaDF/" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 39 "It can be done in a more efficient wa
y:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "x:=array(1..3,[seq(u[j][3][
1],j=1..3)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'vectorG6#7%-
F&6#7%$\"+%4B-E$!#5$\"+Qm9A8F.$\"+4()*=2\"!\"*-F&6#7%,&$!+JXhkQF.\"\"
\"%\"IG$!+G-h$3)F.,&$!+veW<5F3F:F;$\"+$>_z(oF.,&$\"+ee+L$)F.F:F;$!+Qgq
aDF.-F&6#7%,&F8F:F;$\"+G-h$3)F.,&F?F:F;$!+$>_z(oF.,&FDF:F;$\"+QgqaDF.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "P:=augment(x[1],x[2],x[
3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'matrixG6#7%7%$\"+%4B-
E$!#5,&$!+JXhkQF,\"\"\"%\"IG$!+G-h$3)F,,&F.F0F1$\"+G-h$3)F,7%$\"+Qm9A8
F,,&$!+veW<5!\"*F0F1$\"+$>_z(oF,,&F;F0F1$!+$>_z(oF,7%$\"+4()*=2\"F=,&$
\"+ee+L$)F,F0F1$!+QgqaDF,,&FGF0F1$\"+QgqaDF," }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "s1:=u[1][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#s1G$\"+.OUj)*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s2:=
u[2][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s2G,&$!+S!=rJ%!#6\"\"\"
%\"IG$\"+TJ;')G!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s3:=u
[3][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s3G,&$!+S!=rJ%!#6\"\"\"%
\"IG$!+TJ;')G!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "E:=matr
ix(3,3,[s1,0,0,0,s2,0,0,0,s3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"EG-%'matrixG6#7%7%$\"+.OUj)*!#5\"\"!F-7%F-,&$!+S!=rJ%!#6\"\"\"%\"IG$
\"+TJ;')GF,F-7%F-F-,&F0F3F4$!+TJ;')GF," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The diagonalization of A \+
is     A=P*E*inverse(P)" }}{PARA 0 "" 0 "" {TEXT -1 19 "We can verify \+
that:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "multiply(P,E,inverse(P)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%$!#G!#6$\"#!*F*$
\"+9+++I!#57%$\"+3+++SF/$!#AF*$!#IF*7%$!\"'F/$\"+3+++qF/$\"+,+++!*F/" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 
"So, it is correct diagonalization." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "The Fibonacci sequence:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=matrix(2,2,[1,1,1,0]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"\"F*7$F*\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "u:=eigenvects(A);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"uG6$7%,&#\"\"\"\"\"#F)*$-%%sqrtG6#\"\"&
\"\"\"F(F)<#-%'vectorG6#7$F'F)7%,&F(F)F+#!\"\"F*F)<#-F36#7$F7F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "x:=array(1..2,[seq(u[j][3][1
],j=1..2)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'vectorG6#7$-F
&6#7$,&#\"\"\"\"\"#F.*$-%%sqrtG6#\"\"&\"\"\"F-F.-F&6#7$,&F-F.F0#!\"\"F
/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "P:=augment(x[1],x[2]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'matrixG6#7$7$,&#\"\"\"
\"\"#F,*$-%%sqrtG6#\"\"&\"\"\"F+,&F+F,F.#!\"\"F-7$F,F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "E:=matrix(2,2,[(1+sqrt(5))/2,0,0,(1
-sqrt(5))/2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'matrixG6#7$
7$,&#\"\"\"\"\"#F,*$-%%sqrtG6#\"\"&\"\"\"F+\"\"!7$F4,&F+F,F.#!\"\"F-" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Q:=inverse(P);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"QG-%'matrixG6#7$7$,$*$-%%sqrtG6#\"\"&\"
\"\"#\"\"\"F/,$*&,&!\"\"F2F+F2F2F,F0#F2\"#57$,$F+#F6F/,$*&,&F2F2F+F2F2
F,F0F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Verify the diagonalizat
ion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "simplify(multiply(P,E,Q));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#7$7$\"\"\"F(7$F(\"\"!" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Example: Solve \+
the difference equation  " }{XPPEDIT 18 0 "alpha[k] = 5*alpha[k-1]/6-1
*alpha[k-2]/6;" "6#/&%&alphaG6#%\"kG,&*&*&\"\"&\"\"\"&F%6#,&F'F,\"\"\"
!\"\"F,F,\"\"'F1F,*&*&\"\"\"F,&F%6#,&F'F,\"\"#F1F,F,\"\"'F1F1" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A:=matrix(2,2,[5/6,-1/6
,1,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$#\"\"
&\"\"'#!\"\"F,7$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "u:=eigenvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG6$7%#\"
\"\"\"\"$F(<#-%'vectorG6#7$F(F)7%#F(\"\"#F(<#-F,6#7$F(F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "P:=matrix(2,2,[1,1,3,2]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'matrixG6#7$7$\"\"\"F*7$\"\"$\"\"#" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "E:=matrix(2,2,[1/3,0,0,1/
2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'matrixG6#7$7$#\"\"\"
\"\"$\"\"!7$F-#F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Q
:=inverse(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG-%'matrixG6#7$7
$!\"#\"\"\"7$\"\"$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Verify:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "multiply(P,E,Q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%'matrixG6#7$7$#\"\"&\"\"'#!\"\"F*7$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 45 "Example:    Solve the difference equation    " }{XPPEDIT 18 0 "
beta[k] = 5*beta[k-1]/6+beta[k-2]/3-beta[k-3]/6;" "6#/&%%betaG6#%\"kG,
(*&*&\"\"&\"\"\"&F%6#,&F'F,\"\"\"!\"\"F,F,\"\"'F1F,*&&F%6#,&F'F,\"\"#F
1F,\"\"$F1F,*&&F%6#,&F'F,\"\"$F1F,\"\"'F1F1" }{TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 42 "A:=matrix(3,3,[5/6,1/3,-1/6,1,0,0,0,1,0])
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7%7%#\"\"&\"\"'
#\"\"\"\"\"$#!\"\"F,7%F.\"\"!F37%F3F.F3" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "u:=eigenvects(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"uG6%7%#\"\"\"\"\"$F(<#-%'vectorG6#7%F(F)\"\"*7%F(F(<#-F,6#7%F(F(F(
7%#!\"\"\"\"#F(<#-F,6#7%F(!\"#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "x:=array(1..3,[seq(u[j][3][1],j=1..3)]);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"xG-%'vectorG6#7%-F&6#7%\"\"\"\"\"$\"\"*-F&6
#7%F,F,F,-F&6#7%F,!\"#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "P:=augment(x[1],x[2],x[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"PG-%'matrixG6#7%7%\"\"\"F*F*7%\"\"$F*!\"#7%\"\"*F*\"\"%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "E:=matrix(3,3,[1/3,0,0,0,1,0,0,0,-1
/2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG-%'matrixG6#7%7%#\"\"\"
\"\"$\"\"!F-7%F-F+F-7%F-F-#!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "Q:=inverse(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"QG-%'matrixG6#7%7%#!\"\"\"\"&#\"\"\"\"#5F-7%F.#F.\"\"'#F+F27%#F.F,#!
\"%\"#:#F.F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "multiply(P,
E,Q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%#\"\"&\"\"'#
\"\"\"\"\"$#!\"\"F*7%F,\"\"!F17%F1F,F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "evalm(P),evalm(E),evalm(Q);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%-%'matrixG6#7%7%\"\"\"F(F(7%\"\"$F(!\"#7%\"\"*F(\"\"%-F
$6#7%7%#F(F*\"\"!F47%F4F(F47%F4F4#!\"\"\"\"#-F$6#7%7%#F8\"\"&#F(\"#5F@
7%F(#F(\"\"'#F8FD7%#F(F?#!\"%\"#:#F(FJ" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "x[2]:=vector([1,1,1]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%\"xG6#\"\"#-%'vectorG6#7%\"\"\"F,F," }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "y[2]:=multiply(Q,x[2]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"yG6#\"\"#-%'vectorG6#7%\"\"!\"\"\"F," }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "51 0 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 }