next up previous
Next: About this document ...

鐃緒申鐃緒申鐃緒申鐃緒申鐃� I 鐃緒申鐃銃ワ申鐃夙駕申鐃緒申鐃夙駕申鐃緒申

鐃緒申鐃緒申 6.1   $A,E \in M_3({\Bbb C})$, $B\in M_9({\Bbb C})$ 鐃緒申

$A= \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$, $E= \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$, $B= \begin{pmatrix} A & A-E & 0 \\ 0 & A & A-E \\ 0 & 0 & A \end{pmatrix}$鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $\omega=\frac{-1+\sqrt{-3}}{2}$ 鐃夙わ申鐃緒申鐃緒申 鐃緒申鐃塾とわ申鐃緒申

1.
$A^3$ 鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申: $E$. (鐃緒申鐃緒申鐃獣縁申鐃竣誌申鐃叔わ申鐃暑が鐃緒申 鐃緒申鐃旬ベワ申鐃夙ワ申旅圓鐃緒申鐃� ( $e_1\mapsto e_3 \mapsto e_2 \mapsto e_1$)鐃緒申 鐃緒申鐃銃わ申鐃緒申鐃緒申鐃緒申禊鰹申瓩わ申鐃�)

2.
$A$ 鐃塾醐申有鐃粛をすべて居申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申: $1,\omega, \omega^2$. (鐃緒申鐃緒申鐃緒申単鐃緒申鐃緒申鐃緒申鐃緒申鐃淑わ申鐃緒申鐃緒申鐃� 1.鐃塾件申未鐃緒申蕁�$A$ 鐃塾醐申有鐃粛わ申必鐃緒申 3鐃処し鐃緒申 $1$ 鐃祝なるこ鐃夙わ申錣�鐃獣てわ申鐃緒申呂鐃緒申任鐃緒申襦�)

3.
$Q^{-1}AQ$ 鐃緒申鐃出角刻申鐃緒申任鐃緒申鐃処う鐃緒申鐃緒申則鐃緒申鐃緒申 $Q$ 鐃緒申鐃縦居申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申:

\begin{displaymath}Q= \begin{pmatrix} 1 & 1 & 1\\ 1 & \omega & \omega^2 \\ 1 & \omega^2 & \omega \end{pmatrix}\end{displaymath}

( $1,\omega, \omega^2$ 鐃緒申鐃出わ申鐃緒申鐃粛�鐃駿ワ申鐃夙ワ申鐃緒申鐃緒申鐃渋べわ申个茲わ申鐃緒申鐃緒申両鐃緒申砲鐃� $A-E$, $A-\omega E$, $A-\omega^2 E$ 鐃塾核わ申鐃緒申鐃塾わ申鐃緒申鐃緒申鐃潤い鐃緒申鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃駿の刻申鐃緒申覆鐃緒申鐃旬ベワ申鐃夙ワ申旅圓鐃緒申鐃緒申茲�鐃緒申鐃緒申鐃� 鐃緒申鐃緒申鐃緒申鐃旬のわ申鐃叔醐申有鐃駿ワ申鐃夙ルが鐃緒か鐃緒申呂鐃緒申任鐃緒申襦� 鐃淑わ申鐃緒申$Q^{-1}AQ$ 鐃緒申

\begin{displaymath}D= \begin{pmatrix} 1 & 0 & 0 \\ 0 & \omega & 0 \\ 0 & 0 & \omega^2 \end{pmatrix}\end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申)

4.
鐃緒申鐃� $Q$ 鐃祝わ申鐃緒申鐃緒申 $S= \begin{pmatrix} Q & 0 & 0 \\ 0 & Q & 0 \\ 0 & 0 & Q \end{pmatrix}$鐃夙わ申鐃緒申鐃緒申鐃緒申鐃塾とわ申 $S^{-1}BS$ 鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申:


\begin{displaymath}S^{-1}BS = \begin{pmatrix} D & D-E & 0 \\ 0 & D & D-E \\ 0 ... ...1 & 0 & 0 \\ 0 & \omega & 0 \\ 0 & 0 & \omega^2 \end{pmatrix}\end{displaymath}

鐃夙書いてもい鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申分鐃緒申

\begin{displaymath}S^{-1}BS = \begin{pmatrix} 1 & 0 & 0 &0 & 0 & 0& 0 & 0 & 0 \\... ... & 0\\ 0 & 0 & 0& 0 & 0 & 0 &0 & 0 & \omega^2\\ \end{pmatrix}\end{displaymath}

鐃夙書いてわ申茲わ申鐃緒申泙鐃緒申鐃緒申鐃緒申里茲�鐃祝ブワ申鐃獣ワ申鐃緒申媼鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申任盥緒申鐃淑わ申鐃緒申

\begin{displaymath}S^{-1}BS = \begin{pmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & ... ... 0 & \omega & 0 \\ 0 & 0 & \omega^2 \end{bmatrix}\end{pmatrix}\end{displaymath}

5.
$B$ 鐃塾醐申有鐃粛をすべて居申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申: $1,1,1,\omega,\omega,\omega,\omega^2,\omega^2,\omega^2$.

($B$ 鐃塾醐申有鐃緒申鐃緒申鐃緒申(鐃緒申有鐃緒申)鐃緒申 $S^{-1}BS$ 鐃塾醐申有鐃緒申鐃緒申鐃緒申(鐃緒申有鐃緒申)鐃熟わ申鐃淑わ申鐃緒申)

6.
$C=R^{-1}BR$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃宿醐申犒随申砲覆鐃処う鐃緒申 $R,C$ 鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申:

\begin{displaymath}R= \begin{pmatrix} 1 & 0 & 0 & - 3 \omega & 0 & 0 & 3 \omega ... ... 0 & 0 & 1 & 0 & 0 & - \omega -1 & 0 & 0 & \omega \end{pmatrix}\end{displaymath}


\begin{displaymath}C=\begin{pmatrix} E & 0 & 0 \\ 0 & J & 0 \\ 0 & 0 & \bar{J}... ...& 1 & 0\\ 0 & \omega^2 & 1\\ 0 & 0 & \omega^2 \end{pmatrix}) \end{displaymath}

鐃殉わ申 $X=S^{-1}BS$ 鐃緒申鐃緒申戰鐃緒申肇鐃緒申鐃緒申鐃旬わ申鐃暑。

\begin{displaymath}X e_1=e_1 , X e_4=e_4, X e_7=e_7 \end{displaymath}

鐃叔わ申鐃暑か鐃初、$X$ 鐃塾醐申有鐃緒申 $1$ 鐃緒申鐃出縁申鐃緒申鐃緒申鐃粛�鐃駿ワ申鐃夙ワ申箸鐃緒申鐃� $e_1,e_4,e_7$ 鐃緒申鐃緒申鐃暑こ鐃夙わ申鐃緒か鐃暑。同鐃粛にわ申鐃銃￥申 $e_2,e_5,e_8$ 鐃緒申 $X$ 鐃塾醐申有鐃緒申 $\omega$ 鐃緒申鐃出縁申鐃緒申鐃緒申鐃緒申有鐃駿ワ申鐃夙ワ申, $e_3,e_6,e_9$ 鐃緒申 $X$ 鐃塾醐申有鐃緒申 $\omega^2$ 鐃緒申鐃出縁申鐃緒申鐃緒申鐃緒申有鐃駿ワ申鐃夙ワ申任鐃緒申襪鰹申箸鐃緒申鐃緒申鐃緒申砲錣�鐃暑。 鐃緒申鐃塾わ申鐃夙わ申鐃緒申鐃旬わ申鐃銃￥申鐃述とまわ申 $X$ 鐃薯次のよう鐃緒申鐃術器申鐃緒申鐃暑。

\begin{displaymath}Y=P^{-1}XP= \begin{pmatrix} E & 0 & 0\\ 0 & L & 0 \\ 0 & 0 & \bar{L} \end{pmatrix}\end{displaymath}

鐃緒申鐃緒申鐃緒申 $P$ 鐃熟種申鐃塾よう鐃淑器申鐃緒申鐃緒申峇鐃緒申鐃緒申弍鐃緒申鐃緒申鐃緒申鐃緒申任鐃緒申襦�

\begin{displaymath}\begin{pmatrix} e_1 &e_2& e_3& e_4& e_5& e_6& e_7& e_8& e_9\\... ...1 & e_4 & e_7 & e_2 & e_5 & e_8 & e_3 & e_6 & e_9 \end{pmatrix}\end{displaymath}

鐃緒申鐃緒申 $P$ 鐃緒申 $(1,1),(4,2),(7,3),(2,4),(5,5),(8,6),(3,7),(6,8),(9,9)$ 鐃塾鰹申鐃緒申分 鐃緒申 $1$ 鐃叔残わ申鐃� $0$ 鐃叔わ申鐃緒申茲�鐃淑刻申鐃緒申任鐃緒申襦� $P$ 鐃緒申鐃塾わ申颪�のはばわ申鐃出わ申鐃緒申鐃緒申鐃塾でわ申鐃緒申鐃叔は割愛鐃緒申鐃暑。


\begin{displaymath}L=\begin{pmatrix} \omega &\omega-1 & 0 \\ 0 & \omega & \omeg... ...\\ 0 & \omega^2 & \omega^2-1\\ 0 & 0 & \omega^2 \end{pmatrix}\end{displaymath}

鐃叔わ申鐃暑。

鐃緒申鐃夙わ申 $\omega-1$, $\omega^2-1$ 鐃緒申鐃緒申分鐃緒申鐃緒申鐃緒申鐃緒申襪鰹申箸鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃獣縁申縫戰鐃緒申肇鐃緒申 鐃緒申鐃旬わ申鐃緒申鐃緒申个茲わ申鐃�

鐃緒申鐃緒申鐃緒申鐃塾件申鐃叔述べてわ申鐃緒申鐃夙つわ申鐃塾よう鐃祝なる。

鐃緒申鐃緒申 6.1  

\begin{displaymath}X=\begin{pmatrix} a & b & 0 \\ 0 & a & c \\ 0 & 0 & a \end{pmatrix}\end{displaymath}

($b\neq 0$, $c\neq 0$) 鐃塾ワ申鐃緒申鐃緒申鐃緒申鐃宿醐申犒随申鐃�

\begin{displaymath}Y^{-1}XY= \begin{pmatrix} a & 1 & 0 \\ 0 & a & 1 \\ 0 & 0 & a \end{pmatrix}\end{displaymath}

鐃叔わ申鐃緒申鐃緒申鐃緒申鐃暑。鐃緒申鐃緒申鐃叔￥申$Y$ 鐃緒申

\begin{displaymath}Y= \begin{pmatrix} bc & 0 & 0 \\ 0 & c & 0 \\ 0 & 0 & 1 \end{pmatrix}\end{displaymath}

鐃緒申与鐃緒申鐃緒申鐃緒申鐃緒申鐃叔わ申鐃暑。

(鐃淑わ申鐃緒申鐃緒申鐃緒申離鐃緒申鐃緒申鐃緒申鬚�わ申鐃銃わ申同鐃粛のわ申鐃夙をいわ申鐃緒申鐃夙わ申鐃叔わ申鐃暑。)

7.
$B$ 鐃塾ワ申鐃緒申鐃緒申鐃緒申分鐃緒申鬚靴覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申:

\begin{displaymath}B=Z+N \end{displaymath}


\begin{displaymath}Z= \begin{pmatrix} A & 0 & 0 \\ 0 & A & 0 \\ 0 & 0 & A \end{pmatrix}\end{displaymath}


\begin{displaymath}N= \begin{pmatrix} 0 & A-E & 0 \\ 0 & 0 & A-E \\ 0 & 0 & 0 \end{pmatrix}\end{displaymath}

(鐃緒申鐃曙が $B$ 鐃塾ワ申鐃緒申鐃緒申鐃緒申分鐃緒申鐃粛随申鐃緒申襪鰹申箸蓮鐃緒申鐃緒申鐃�5.1 鐃緒申1.-4. 鐃殉での緒申鐃緒申鐃緒申鐃緒申 調鐃駿わ申个茲わ申鐃緒申鐃緒申鐃熟やさ鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申5.1 鐃塾逸申鐃緒申鐃緒申琉鐃緒申呂任鐃緒申襦�)

鐃緒申鐃緒申 6.2   $A,E \in M_3({\Bbb C})$, $B\in M_9({\Bbb C})$ 鐃緒申

$A= \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 2 & 0 & 0 \end{pmatrix}$, $E= \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$, $B= \begin{pmatrix} A & A-E & 0 \\ 0 & A & A-E \\ 0 & 0 & A \end{pmatrix}$鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $\omega=\frac{-1+\sqrt{-3}}{2}$ 鐃夙わ申鐃緒申鐃緒申 鐃緒申鐃塾とわ申鐃緒申

1.
$A^3$ 鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申:$2E$.

2.
$A$ 鐃塾醐申有鐃粛をすべて居申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申: $\sqrt[3]{2},\sqrt[3]{2}\omega,\sqrt[3]{2}\omega^2$.

3.
$Q^{-1}AQ$ 鐃緒申鐃出角刻申鐃緒申任鐃緒申鐃処う鐃緒申鐃緒申則鐃緒申鐃緒申 $Q$ 鐃緒申鐃縦居申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申:

\begin{displaymath}Q=\begin{pmatrix} 1 & 1 & 1 \\ {\sqrt[3]{2}} & {\sqrt[3]{2}}... ...4}} & {\sqrt[3]{4}}\omega^2 & {\sqrt[3]{4}}\omega \end{pmatrix}\end{displaymath}

4.
鐃緒申鐃� $Q$ 鐃祝わ申鐃緒申鐃緒申 $S= \begin{pmatrix} Q & 0 & 0 \\ 0 & Q & 0 \\ 0 & 0 & Q \end{pmatrix}$鐃夙わ申鐃緒申鐃緒申鐃緒申鐃塾とわ申 $S^{-1}BS$ 鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申:

\begin{displaymath}S^{-1}BS = \begin{pmatrix} D & D-E & 0 \\ 0 & D & D-E \\ 0 ... ...]{2}}\omega & 0 \\ 0 & 0 & {\sqrt[3]{2}}\omega^2 \end{pmatrix}\end{displaymath}

5.
$B$ 鐃塾醐申有鐃粛をすべて居申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申: $\sqrt[3]{2},\sqrt[3]{2}\omega,\sqrt[3]{2}\omega^2$ (鐃緒申鐃曙ぞ鐃緒申3鐃緒申)

6.
$C=R^{-1}BR$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃宿醐申犒随申砲覆鐃処う鐃緒申 $R,C$ 鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申: 鐃緒申鐃緒申1 鐃緒申与鐃緒申鐃緒申鐃塾とわ申鐃淑わ申 $P$ 鐃夙常申鐃� $S$ 鐃夙わ申鐃術わ申鐃銃￥申 $P^{-1}S^{-1}BSP$鐃緒申彁鐃緒申鐃緒申討澆鐃夙￥申

\begin{displaymath}\begin{pmatrix} {\sqrt[3]{2}} & {\sqrt[3]{2}} -1 & 0 & 0 & 0 ... ...0 & 0 & 0 & 0 & 0 & 0 & {\sqrt[3]{2}} \omega^2\\ \end{pmatrix}\end{displaymath}

鐃夙なる。鐃出角わ申鐃渋ぶ誌申鐃縦のブワ申鐃獣ワ申鐃緒申鐃緒申鐃縦ともこ鐃緒申幣鐃瞬ワ申鐃獣ワ申分鐃緒申 鐃叔わ申鐃淑わ申鐃緒申鐃薯してわ申鐃暑。 鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申1 鐃緒申鐃順き鐃緒申鐃循なわ申箸鐃緒申鐃緒申任鐃緒申襦�

鐃緒申鐃夙わ申鐃緒申鐃緒申1鐃緒申同鐃粛わ申鐃出角刻申鐃緒申之鐃緒申鐃緒申鐃緒申鐃緒申鐃出よい鐃緒申 $R$ 鐃熟種申鐃塾よう鐃祝なる。(dvi鐃春ワ申鐃緒申鐃緒申鐃叔でわ申鐃順き鐃緒申鐃緒申鐃緒申鐃緒申鐃塾を見るこ鐃夙わ申鐃叔わ申鐃淑わ申鐃塾で￥申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申html鐃春ワ申鐃緒申鐃緒申鐃叔のわ申里鮓�わ申里鬚�器申鐃潤す鐃暑。


\begin{displaymath}\begin{pmatrix} -2{\sqrt[3]{2}} + {\sqrt[3]{4}} + 1 & 0 & 0 ... ...{4}} \omega^2 & 0 & 0 & {\sqrt[3]{4}} \omega \\ \end{pmatrix}\end{displaymath}

$C$ 鐃熟種申鐃塾よう鐃祝なる。


\begin{displaymath}\begin{pmatrix} {\sqrt[3]{2}} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & ... ... & 0 & 0 & 0 & 0 & 0 & 0 & {\sqrt[3]{2}} \omega^2 \end{pmatrix}\end{displaymath}

鐃緒申鐃緒申魯鐃楯￥申鐃緒申鐃緒申鐃緒申罎�鐃緒申鐃緒申鐃祝誌申鐃祝わ申鐃銃みわ申鐃緒申鐃緒申鐃緒申辰鐃緒申戮鐃緒申瓩�鐃緒申鐃緒申鐃緒申鐃塾わ申覆鐃緒申鐃�

7.
$B$ 鐃塾ワ申鐃緒申鐃緒申鐃緒申分鐃緒申鬚靴覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申:

\begin{displaymath}B=Z+N \end{displaymath}


\begin{displaymath}Z= \begin{pmatrix} A & 0 & 0 \\ 0 & A & 0 \\ 0 & 0 & A \end{pmatrix}\end{displaymath}


\begin{displaymath}N= \begin{pmatrix} 0 & A-E & 0 \\ 0 & 0 & A-E \\ 0 & 0 & 0 \end{pmatrix}\end{displaymath}

next up previous
Next: About this document ...

2001-01-31