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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.3

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II・Cayley Hamilton の定理}

鐃緒申鐃緒申癲�$k$ 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。

鐃緒申鐃藷、種申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃縦わ申鐃緒申鐃緒申

鐃緒申鐃緒申 1 (鐃緒申鐃緒申鐃緒申鐃緒申鐃�2.5)   $A\in M_n(k)$ 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緒申 鐃緒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� 1 (鐃緒申鐃緒申鐃緒申鐃緒申2.3)   $V_i$ 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 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 3.1   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 & 9 & 8\\ -1 & 1 & 7& 6\\ 0 & 0& 5 & 2 \\ 0 & 0 & -3 & 0 \end{pmatrix}\end{displaymath}

2.

\begin{displaymath}\begin{pmatrix} 3 & 1 & 1 & 2 \\ -1 & 1 &3 & 4\\ 0 & 0 & 3 & 1 \\ 0 & 0 & -1 & 1 \end{pmatrix}\end{displaymath}


Yoshifumi Tsuchimoto
2000-10-21