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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.2

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{行列の固有値と弱固有空間}

鐃緒申鐃緒申蓮鐃�$k$ 鐃夙わ申鐃緒申鐃緒申鐃塾わ申悗鐃緒申鐃塾とわ申鐃暑。

鐃緒申鐃� 1   $M_n(k)$ 鐃塾醐申 $A$ 鐃緒申与鐃緒申鐃緒申譴随申箸鐃緒申襦� 鐃盾し $A$ 鐃塾最常申多鐃準式鐃夙は￥申$f(A)=0$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃処う鐃緒申 $f\in k[X]\setminus \{0\}$ 鐃塾わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃叔常申鐃塾わ申里里鐃緒申箸任鐃緒申襦�

鐃緒申鐃緒申 1   $M_n(k)$ 鐃塾醐申 $A$ 鐃緒申与鐃緒申鐃緒申譴随申箸鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃�
1.
$A$ 鐃塾最常申多鐃準式 $f$ 鐃緒申鐃緒申鐃緒申椶鐃緒申鐃緒申鐃銃逸申佞鐃渋醐申澆鐃緒申襦�
2.
鐃縦とわ申鐃銃わ申同鐃緒申 $k[A]\isoto k[X]/(f)$ 鐃緒申鐃淑りた鐃縦￥申

鐃緒申鐃緒申 2   $M_n(k)$ 鐃塾醐申 $A$ 鐃緒申与鐃緒申鐃緒申譴随申箸鐃緒申鐃緒申鐃緒申虜脳鐃渋随申犲逸申鐃� $f$ 鐃夙わ申鐃暑。 鐃緒申鐃塾とわ申鐃緒申任鐃春わ申 $a\in k$ 鐃緒申鐃出わ申鐃銃￥申

\begin{displaymath}\text{$A-aE $ は可逆行列}\ \iff\ f(a)\neq 0 \end{displaymath}

鐃緒申鐃淑りた鐃縦￥申

鐃緒申鐃� 2   $A$ 鐃塾最常申多鐃準式 $f$ 鐃塾削申(鐃緒申複鐃熟刻申慮鐃緒申鐃淑わ申)鐃緒申 $f$ 鐃塾醐申有鐃粛と呼ぶ￥申

(鐃緒申鐃緒申) 鐃緒申複鐃緒申鐃塾醐申鐃緒申鐃緒申鐃粛�鐃粛わ申鐃緒申鐃緒申砲麓鐃緒申鐃� Cayley-Hamilton 鐃緒申鐃緒申鐃緒申 鐃緒申必鐃竣になる。

鐃緒申鐃緒申 1 (Cayley-Hamilton)   $n$-鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $A$ 鐃緒申鐃出わ申鐃銃￥申鐃緒申鐃塾醐申有多鐃準式 $\Phi_A$ 鐃緒申

\begin{displaymath}\Phi_A(X)=\det(XE-A) \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃� $\Phi_A(A)=0$ 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祝つわ申鐃銃は種申鐃緒申暴劼戮襦ｏ申鐃緒申鐃緒申鐃緒申鐃緒申辰萄鐃緒申鐃�(鐃塾ワ申檗鐃緒申鐃緒申鐃緒申鐃�) 鐃熟わ申鐃緒申鐃緒申鐃緒申鐃緒申眺鐃緒申鐃� $A$ 鐃塾最常申多鐃準式鐃塾醐申鐃緒申鐃緒申佞鐃緒申襪鰹申箸鐃緒申鐃緒申鐃緒申鐃渋�鐃緒申鐃銃わ申鐃緒申 鐃緒申鐃夙にわ申鐃暑。

直鐃緒申分鐃緒申閥鐃緒申鐃緒申鐃緒申鐃塾関件申鐃祝つわ申鐃緒申鐃緒申鐃緒申鐃緒申鐃銃わ申鐃緒申鐃緒申鐃緒申

鐃緒申鐃緒申 3   $P\in M_n(k)$ 鐃緒申鐃緒申鐃緒申($P^2=P$) 鐃淑わ申弌鐃緒申鐃緒申里鐃緒申箸鐃緒申鐃緒申鐃塾�鐃縦￥申
1.
$Q=E-P$ 鐃緒申鐃緒申鐃緒申任鐃緒申蝓�

\begin{displaymath}PQ=0, QP=0, P^2=P, Q^2=Q, P+Q=E \end{displaymath}

鐃緒申鐃淑りた鐃縦￥申
2.
$k^n$ 鐃塾醐申 $v$ 鐃緒申

\begin{displaymath}v=v_1+v_2 \quad (v_1\in \Image(P) , v_2\in \Image(Q)) \end{displaymath}

鐃夙逸申鐃重�鐃祝書ける。(鐃緒申鐃淑わ申鐃緒申鐃�$k^n$ 鐃緒申 $\Image(P)$ 鐃緒申 $\Image(Q)$ 鐃夙わ申 直鐃渋でわ申鐃暑。

鐃緒申鐃緒申 4   任鐃春わ申 $A\in M_n(k)$ 鐃緒申任鐃春わ申 $a\in k$ 鐃緒申鐃出わ申鐃銃種申鐃塾よう鐃緒申 $k^n$ 鐃緒申直鐃緒申分鐃緒申 $k^n =V\oplus W $鐃緒申存鐃淳わ申鐃緒申
1.
$AV \subset V$, $AW \subset W$.
2.
$V$ 鐃緒申任鐃� $A-aE $ 鐃熟駈申鐃緒申 $($鐃縦まり、$V$ 鐃緒申鐃�$A$ 鐃塾最常申多鐃準式鐃緒申 $(X-a)^l$ $(\exists l)$.$)$
3.
$W$ 鐃緒申任鐃� $A$ 鐃熟可居申 $($鐃縦まり、$W$ 鐃緒申鐃� $A$ 鐃塾最常申多鐃準式鐃緒申 $a$ 鐃緒申鐃祝もた鐃淑わ申$)$

鐃緒申鐃緒申 5   $A\in M_n(k)$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃銃￥申鐃緒申鐃塾最常申多鐃準式 $f$ 鐃緒申 $k$ 鐃緒申鐃渋随申犲逸申箸鐃緒申鐃� 鐃曙次鐃緒申鐃緒申鐃術わ申分鐃緒申任鐃緒申鐃�

\begin{displaymath}f(X)= (X-\lambda_1)^{e_1} (X-\lambda_2)^{e_2} (X-\lambda_3)^{e_3} \dots (X-\lambda_l)^{e_l} \end{displaymath}

鐃夙わ申鐃暑。鐃緒申鐃塾とわ申鐃緒申 $k^n$ 鐃緒申直鐃緒申分鐃緒申

\begin{displaymath}k^n =V_1\oplus V_2\oplus V_3\oplus \dots \oplus V_l \end{displaymath}

鐃叔￥申
1.
$AV_i\subset V_i$
2.
鐃緒申 $V_i$ 鐃緒申任鐃� $A-\lambda_i E $ 鐃熟駈申鐃緒申
鐃夙なわ申鐃塾わ申存鐃淳わ申鐃暑。$V_i$ 鐃熟種申鐃塾緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申
1.
$V_i$ 鐃叔わ申 $A$ 鐃塾最常申多鐃準式鐃熟わ申鐃順う鐃緒申 $(X-\lambda_i)^{e_i}$ 鐃叔わ申鐃暑。
2.
$V_i=\{v\in k^n; A^Nv=0 \quad(\exists N>0) \}$
3.
$V_i=\{v\in k^n; A^{e_i}v=0 \}$
4.
鐃緒申鐃緒申 $p_i\in k[X]$ 鐃緒申鐃緒申鐃獣て￥申 $P_i=p_i(A)$ 鐃熟駈申鐃緒申($P_i^2=P_i$)鐃緒申鐃縦￥申 $V_i=P_i( k^n)$ 鐃緒申鐃淑りた鐃縦￥申

鐃緒申鐃� 3   $V_i$ 鐃塾わ申鐃夙わ申 $A$ 鐃塾醐申有鐃緒申 $\lambda_i$ 鐃緒申属鐃緒申鐃緒申鐃緒申有鐃緒申鐃瞬と呼び￥申 $P_i$ 鐃塾わ申鐃夙わ申 $V_i$ 鐃緒申鐃出縁申鐃緒申鐃緒申猶討噺討屐鐃� 鐃緒申

鐃緒申鐃緒申 2.1   鐃緒申鐃緒申(複鐃叔随申鐃緒申鐃緒申分鐃祝わ申鐃�)鐃銃刻申鐃緒申慮鐃粛�鐃緒申(鐃緒申複鐃熟刻申慮鐃緒申鐃淑わ申)鐃夙￥申 鐃緒申鐃緒申鐃渋逸申鐃緒申鐃緒申鐃粛�鐃緒申鐃瞬わ申鐃出縁申鐃緒申鐃緒申猶討鐃緒申鐃緒申撞鐃緒申茵�
1.

\begin{displaymath}\begin{pmatrix} 3 & 1 \\ -1 & 1 \end{pmatrix}\end{displaymath}

2.

\begin{displaymath}\begin{pmatrix} 5 & 2 \\ -3 & 0 \end{pmatrix}\end{displaymath}

3.

\begin{displaymath}\begin{pmatrix} 3 & 1 & 0 \\ -1 & 1 &0 \\ 0 & 0 & 7 \end{pmatrix}\end{displaymath}

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Yoshifumi Tsuchimoto
2000-10-21