第17章 17.19

$|P| = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{vmatrix} = 1 - 1 - 1 = -1$

$P_{11} = \begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} = 1 \quad P_{12} = \begin{vmatrix} 1 & 0 \\ 1 & 1 \end{vmatrix} = 1 \quad P_{13} = \begin{vmatrix} 1 & 1 \\ 1 & 0 \end{vmatrix} = -1$

$P_{21} = \begin{vmatrix} 1 & 1 \\ 0 & 1 \end{vmatrix} = 1 \quad P_{22} = \begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} = 0 \quad P_{23} = \begin{vmatrix} 1 & 1 \\ 1 & 0 \end{vmatrix} = -1$

$P_{31} = \begin{vmatrix} 1 & 0 \\ 1 & 0 \end{vmatrix} = -1 \quad P_{32} = \begin{vmatrix} 1 & 1 \\ 1 & 0 \end{vmatrix} = -1 \quad P_{33} = \begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} = 0$

$P^{-1} = \begin{pmatrix} -1 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix}$

$P^{-1}AP = \begin{pmatrix} -1 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 1 & 1 \\ 0 & 2 & 1 \\ 0 & 0 & 3 \end{pmatrix} \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix} = \begin{pmatrix} -1 & 1 & 3 \\ 1 & 1 & -2 \\ 1 & -1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 3 & 0 & 2 \\ 0 & 2 & -1 \\ 0 & 0 & 1 \end{pmatrix} = B$

$|B - \lambda E| = \begin{vmatrix} 3-\lambda & 0 & 2 \\ 0 & 2-\lambda & -1 \\ 0 & 0 & 1-\lambda \end{vmatrix} = (3-\lambda)(2-\lambda)(1-\lambda) = 0 \quad \lambda = 1,2,3$

$\lambda = 1$ のとき

$\begin{pmatrix} 2 & 0 & 2 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \end{pmatrix} \to \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \end{pmatrix} \quad p_1 = C_1 \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}$

$\lambda = 2$ のとき

$\begin{pmatrix} 1 & 0 & 2 \\ 0 & 0 & -1 \\ 0 & 0 & -1 \end{pmatrix} \to \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \quad p_2 = C_2 \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$

$\lambda = 3$ のとき

$\begin{pmatrix} 0 & 0 & 2 \\ 0 & -1 & -1 \\ 0 & 0 & -2 \end{pmatrix} \to \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \quad p_3 = C_3 \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$