🚀 go-pugleaf

$B$3$l$J$i;d$K$b$G$-$k$+$b(B...$B$H;W$C$F$d$C$F$_$^$7$?(B
$B$,!$$J$+$J$+Fq$7$$$G$9$M!%(B

$B#D#r#a#m$5$s(B>
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: $B!ZLdBj#2![(B
: $B!!9TNs#A$N%G%#%?!<%_%J%s%H$,#1$G$"$k$H$9$k!#(B
: $B#A$N#3>h$,C10L9TNs$H$J$k$H$-!"#A$N%H%l!<%9$r5a$a$h!#(B
: ----------------------------------------------------------------------

: $B!ZLdBj#2$N;d$NEz0F![(B
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: $B!!#A$N%H%l!<%9$r(Bt$B$H$9$k!#(B
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: $B!!%1!<%j!<%O%_%k%H%s$NK!B'$h$j!"(B
: $B!!!!#A(B^2$B!](Bt$B#A!\#E!a#O!!!!!J#E$OC10L9TNs!$#O$ONm9TNs!K(B
: 
: $B!!$h$C$F!"#A(B^3$B!a!J(Bt^2$B!](B1$B!K#A!](Bt$B#E(B
: 
: $B!!#A(B^3$B!a#E$N$H$-!"(B
: $B!!!!!J(Bt^2$B!](B1$B!K#A!](Bt$B#E!a#E(B
: $B!!!!!J(Bt^2$B!](B1$B!K#A!a!J(B1$B!\(Bt$B!K#E(B
: $B!!!!(B              t$B!\(B1            1
:     $B$h$C$F#A!a(B ---------$B#E!!!a!!(B-----$B#E!!(B(*)
:                  t^2$B!](B1          t-1
: 
: $B!!(B(*)$B$G!"9TNs#A$N%G%#%?!<%_%J%s%H$,#1$H$J$k$N$G!"(B
:                1
: $B!!!!(Bdet$B#A!a!J(B-----$B!K(B^2$B!a#1!!$h$C$F(Bt$B!a#2(B
:               t-1
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: $B!!$f$($K#A$N%H%l!<%9$O!!#2!!!JEz!K(B
: ----------------------------------------------------------------------

$B$3$N2rEz$@$H!$(B

  (1) det$B#A(B=1 $B$K$J$kM}M3$,=q$$$F$"$j$^$;$s!%(B
  (2) (t^2-1) $B$G3d$k$H$3$m$G!$(Bt=1,-1 $B$N;~$H$=$&$G(B
      $BL5$$$H$-$N>l9gJ,$1$r$7$F$$$J$$!%(B
  (3) $B<B:]$K%H%l!<%9$,(B 2 $B$K$J$k$b$N$,B8:_$9$k$H$$(B
      $B$&$3$H$r<($7$F$"$j$^$;$s!%(B

$B$N;0E@$,%^%:%$$H;W$$$^$9!%(B

$BFC$K(B (3) $B$rK:$l$k@8EL$O$+$J$jB?$$$N$G$O$J$$$G$7$g(B
$B$&$+!)(B $B7W;;$J$s$+$h$j$b$3$NJU$j$,Bg;v$J$s$8$c$J$$(B
$B$+$C$F5$$b$7$^$9!%(B

$B;d$N2rEz$@$H!$$A$g$C$HD9$$$G$9$,!$(B

   $B2r(B)

   A $B$O<B?t$r@.J,$H$9$k(B 2x2 $B9TNs$G$"$k$H2>Dj$7$F2r(B
   $B$/!%(B

           1=det(E)=det(A^3)=det(A)^3

   $B$h$j(B

                  det(A) = 1

   t=trace(A) $B$H$*$/$H!$(BCayley=Hamilton $B$NEy<0$h$j(B

              A^2 = tA + det(A)E
                  = tA + E

   $B$3$l$H(B A^3 = E $B$H$+$i!$(B

              (t^2-1)A = (1+t)E
          (1+t){(t-1)A - E} = 0 $B!D(B(*)


   $B0J2<!$(Bt=-1 $B$N$H$-$H(B t$B!b(B1 $B$N$H$-$KJ,$1$F9M$($k(B

   i) t=-1 $B$N>l9g(B

      (*)$B$rK~$9!%<B:]!$(B

               /   -1/2     -(1/2)$B"e(B3 \
           A = |                      |
               \ (1/2)$B"e(B3     -1/2    /

      $B$N$H$-!$(BA^3=E $B$+$D(B traceA = -1

   ii) t$B!b(B-1 $B$N>l9g(B

      (*) $B$h$j!$(B

               (t-1)A - E = 0

      t=1 $B$H$9$k$H$3$N<0$,@.N)$7$J$$$N$G!$(Bt$B!b(B1
      t-1 ($B!b(B0) $B$G3d$C$F!$(B

                   A = E/(t-1)

      trace $B$r7W;;$9$k$H!$(B

             t = trace A = 2/(t-1)
             t^2 - t - 2 = 0
               (t-2)(t+1)= 0

      t$B!b(B-1 $B$h$j!$(B

                   t = 2

      $B<B:]!$(BA=E $B$N$H$-!$(BA^3 = E $B$+$D!$(BtraceA = 2

    $B0J>e$r$^$H$a$k$H!$(B

      ($BEz(B)  t = 2 , -1

    $B"#(B

$B$C$F$J$H$3$m$G$9!%(B

--
SAKURADA, Hideki
E-mail: sakurada@kusm.kyoto-u.ac.jp