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鐃緒申鐃緒申鐃緒申鐃緒申鐃� I 鐃緒申鐃緒申 No.10

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

$\fbox{PID 上の行列のスミス標準型}$

(鐃緒申法)

鐃縦器申鐃緒申鐃祝つわ申鐃銃￥申鐃緒申鐃塾よう鐃淑居申鐃緒申鐃緒申僂鐃緒申襦�

$\begin{displaymath}M_{m,n}(R)=\text{($R$ の元を成分に持つ $m\times n$-行列)} \end{displaymath}$

$\begin{displaymath}M_n(R)=M_{n,n}(R) \quad \text{(正方行列全体: 既出)} \end{displaymath}$

$\begin{displaymath}GL_n(R)=M_n(R)^\times \end{displaymath}$

鐃緒申鐃緒申 10.1

鐃緒申PID鐃叔￥申 $M_{m,n}(R)$ 鐃塾醐申

鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� 鐃緒申鐃塾とわ申鐃緒申 $P\in {\operatorname{GL}}_{m}(R)$ 鐃緒申 $Q\in {\operatorname{GL}}_{n}(R)$ 鐃夙をうまわ申与鐃緒申鐃緒申弌鐃� PAQ 鐃熟種申鐃塾よう鐃淑件申鐃祝でわ申鐃暑。

$\begin{displaymath}PAQ= \left( \begin{array}{c c c c c \vert c c c } e_1 & & & &... ...r1.7ex\hbox{\huge0}}}\\ & & & & & & & \\ \end{array}\right) \end{displaymath}$

( 鐃緒申鐃緒申鐃緒申, $e_1,\dots , e_k$ 鐃緒申鐃塾醐申鐃叔￥申 $e_1 \vert e_2 \vert e_3 \vert\dots\vert e_k$ 鐃緒申鐃淑りた鐃縦￥申)

鐃緒申鐃緒申鐃緒申鐃緒申鐃塾常申鐃緒申鐃緒申鐃旬種申的鐃祝わ申 $2\times 2$ 鐃緒申鐃緒申鐃緒申辰鐃緒申鐃緒申鐃緒申鐃緒申鐃叔わ申鐃暑。

鐃純え鐃緒申 $M_{2,2}({\mbox{${\Bbb Z}$ }})=M_2({\mbox{${\Bbb Z}$ }})$ 鐃塾醐申

$\begin{displaymath}A=\begin{pmatrix} 30 & 42 \\ 55 & 33 \end{pmatrix}\end{displaymath}$

鐃緒申鐃緒申鐃緒申鐃緒申襦ｏ申泙鐃�

鐃緒申

(鐃緒申1鐃緒申)鐃夙でユー鐃緒申鐃緒申奪匹慮濬鐃祝�(gcd 鐃緒申6)鐃緒申圓鐃緒申鐃�

$\begin{displaymath}3\cdot 30-2\cdot 42=6 ,\qquad 3\cdot (30/6)-2\cdot (42/6)=1 \end{displaymath}$

鐃夙わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申(鐃緒申鐃塾種申鐃祝わ申鐃緒申鐃緒申鐃塾は逸申鐃縮わ申PID

鐃叔はワ申鐃叔ワ申鐃緒申竜鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申襦�) 鐃緒申鐃緒申鐃旬の種申鐃緒申鐃緒申

$\begin{displaymath}\begin{pmatrix} 3 & 7\\ -2 & -5 \end{pmatrix}\end{displaymath}$

鐃緒申 ${\operatorname{GL}}_2({\mbox{${\Bbb Z}$ }})$ 鐃塾醐申鐃叔わ申鐃暑こ鐃夙わ申鐃緒か鐃暑。

$\begin{displaymath}A \begin{pmatrix} 3 & 7 \\ -2 & -5 \end{pmatrix}= \begin{pmatrix} 6 & 0\\ 99 & 220 \end{pmatrix}\end{displaymath}$

鐃緒申鐃駿は￥申

鐃緒申

(鐃緒申1鐃緒申)鐃夙わ申同鐃粛なわ申鐃夙わ申圓鐃緒申鐃�

$\begin{displaymath}\begin{pmatrix} 17 & -1 \\ 33 & -2 \end{pmatrix}A \begin{pma... ...d{pmatrix}= \begin{pmatrix} 3 & -220\\ 0 & -440 \end{pmatrix}\end{displaymath}$

鐃緒申鐃駿は再わ申1鐃峻わ申同鐃粛わ申鐃緒申鐃緒申圓鐃緒申鐃� (1,1)-鐃緒申分鐃緒申鐃淑￥申``鐃緒申鐃緒申鐃緒申''鐃淑ってわ申鐃緒申鐃緒申鐃夙わ申鐃緒申鐃春￥申

$\begin{displaymath}\begin{pmatrix} 17 & -1 \\ 33 & -2 \end{pmatrix}A \begin{pma... ...d{pmatrix}= \begin{pmatrix} 1 & 0\\ 440 & -1320 \end{pmatrix}\end{displaymath}$

鐃叔醐申砲發�申鐃緒申鐃緒申鐃緒申鐃緒申砲弔鐃緒申鐃銃縁申鐃緒申鐃緒申箸鬚垢襦�

$\begin{displaymath}\begin{pmatrix} 1 & 0 \\ -440 &1 \end{pmatrix}\begin{pmatri... ...end{pmatrix}= \begin{pmatrix} 1 & 0\\ 0 & -1320 \end{pmatrix}\end{displaymath}$

鐃緒申鼻鐃�

$\begin{displaymath}P= \begin{pmatrix} 17 & -1\\ -7447 & 438 \end{pmatrix},Q= \begin{pmatrix} -226 & 681\\ 151 & - 455 \end{pmatrix}\end{displaymath}$

鐃緒申鐃瞬わ申鐃出よい鐃緒申

鐃緒申鐃緒申鐃塾よう鐃淑計誌申鐃緒申鐃緒申鐃縦でわ申澆泙襪鰹申箸鬚い鐃緒申砲蓮鐃� 鐃緒申PID 鐃塾ワ申鐃叔ワ申鐃緒申鐃緒申鐃緒申鐃緒申鵑�鐃緒申弔任鐃宿わ申鐃緒申鐃叔とまわ申廚箸鐃緒申鐃緒申鐃緒申造鐃� 鐃緒申鐃春わ申鐃緒申个茲わ申鐃緒申鐃緒申了鐃緒申村鐃緒申里鐃緒申鐃緒申I鐃叔わ申鐃叔に出てわ申鐃銃わ申鐃緒申呂鐃緒申任鐃緒申襦� 鐃緒申鐃緒申同鐃緒申鐃緒申鐃夙をすわ申里呂个鐃緒申个鐃緒申鐃緒申鐃緒申里如鐃� 鐃緒申鐃夙わ申鐃夙のわ申鐃緒申鐃粛わ申鐃銃わ申鐃塾誌申鐃渋の鰹申張鐃叔わ申鐃暑、鐃緒申鐃縮のネ￥申鐃緒申鐃緒申鐃縦わ申鐃緒申徴鐃春わ申鐃祝つわ申鐃銃わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃暑。

鐃緒申鐃緒申 10.1

1.

鐃縦器申鐃粛￥申鐃緒申鐃緒申鐃緒申

鐃塾ワ申鐃叔ワ申鐃緒申鐃緒申鐃緒申鐃緒申鐃�

$\begin{displaymath}I_0\supset I_1 \supset I_2 \supset I_3 \supset \dots \end{displaymath}$

鐃緒申必鐃緒申鐃宿わ申鐃緒申鐃叔止まる。鐃緒申鐃淑わ申鐃緒申鐃緒申鐃緒申鐃�

鐃緒申鐃緒申鐃獣て￥申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃�

$\begin{displaymath}I_N= I_{N+1} \supset I_{N+2}=I_{N+3}= \dots \end{displaymath}$

鐃緒申鐃淑りた鐃縦￥申

2.

鐃春に￥申鐃縦器申鐃緒申

鐃緒申鐃緒申

鐃塾ワ申鐃叔ワ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申必鐃緒申鐃宿わ申鐃緒申鐃叔止まわ申鐃� 鐃夙わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃銃ば￥申

鐃熟ネ￥申鐃緒申鐃緒申鐃縦でわ申鐃暑。

鐃緒申鐃緒申 10.1 $R={\Bbb C}[X]$ 鐃夙わ申鐃暑。

鐃緒申旅鐃緒申鐃�

鐃薯下のよう鐃祝件申鐃緒申箸鐃緒申鐃� 鐃緒申鐃緒申10.1 鐃緒申 $e_1,e_2,\dots$ 鐃祝わ申鐃緒申鐃緒申鐃塾をそれぞ鐃緒申鐃緒申茵� (鐃緒申由鐃春わ申鐃薯きわ申鐃緒申判颪い討鐃緒申鐃出￥申鐃夙わ申鐃緒申

鐃緒申鐃緒申鐃宿�申廚呂覆鐃緒申鐃�)

1.

$\begin{displaymath}A_1=\begin{pmatrix} X^3+X-1 & X^2+X+1 \end{pmatrix}\in M_{1,2}(R) \end{displaymath}$

2.

$\begin{displaymath}A_2=\begin{pmatrix} X & 0 \\ X^5 & 1 \end{pmatrix}\in M_{2,2}(R) \end{displaymath}$

3.

$\begin{displaymath}A_3=\begin{pmatrix} 1 & 1 & 1 \\ X & (X+1) & (X-1) \\ X^2 & (X+1)^2 & (X-1)^2 \end{pmatrix}\in M_{3,3}(R) \end{displaymath}$

(鐃述ワ申鐃�: $\operatorname{det}(A_3)$ 鐃熟わ申鐃緒申鐃初か鐃緒申)

4.

$\begin{displaymath}A_4= \begin{pmatrix} X & X & X^2\\ X^2-X & 2 X^2 & X^3-X^2\\ X^2 & X^2 & X^3 \end{pmatrix}\in M_{3,3}(R) \end{displaymath}$

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Yoshifumi Tsuchimoto
2000-12-04