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�� I �� No.3

��Υơ��:

$\fbox{��θ�ͭ�ͤȼ��ͭ��II��Cayley Hamilton ��}$

��⡢ �Ȥ��Τ�ؤ��ΤȤ��롣

��󡢼��Ĥ��

�� 1 (��2.5) $A\in M_n(k)$ ��Ϳ��Ƥ��ơ��κǾ�¿�༰

��

��¿�༰�Ȥ�� 켡�� ʬ��Ǥ��

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

�Ȥ��롣��ΤȤ��

��ľ��ʬ��

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

�ǡ�

1.: $AV_i\subset V_i$
2.: �� Ǥ� $A-\lambda_i E$ �϶��

�Ȥʤ��Τ�¸�ߤ��롣

�ϼ��ν��

1.: �Ǥ� �κǾ�¿�༰�Ϥ��礦�� $(X-\lambda_i)^{e_i}$ �Ǥ��롣
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.: �� $p_i\in k[X]$ ��äơ� �϶��()��ġ� ��ʤꤿ�ġ�

�� 1 (��2.3)

�Τ��Ȥ�

�θ�ͭ�� $\lambda_i$ ��°��ͭ��֤ȸƤӡ�

�Τ��Ȥ�

��б��ͱƤȸƤ֡� ��

�� 1 (Cayley-Hamilton)

-��

��Ф��ơ��θ�ͭ¿�༰ $\Phi_A$ ��

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

��롣��ΤȤ�� $\Phi_A(A)=0$ ��ʤꤿ�ġ�

�� 3.1 ��(ʣ�ǿ��ʬ�ˤ��)�ƹ��θ�ͭ��(��ʣ�Ϲ�θ��ʤ�)�ȡ� ��°��ͭ��֤��б��ͱƤ��Ƶ��衣

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