第17章 17.16

(1)

$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ として

$\begin{pmatrix} 16 \\ 13 \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 5 \\ 5 \end{pmatrix}$ $\begin{pmatrix} 50 \\ 35 \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 16 \\ 13 \end{pmatrix}$

$\begin{cases} 5a + 5b = 16 \\ 16a + 13b = 50 \end{cases}$ $\begin{cases} 5c + 5d = 13 \\ 16c + 13d = 35 \end{cases}$

これから $a = \frac{14}{5}$ $b = \frac{2}{5}$ $c = \frac{2}{5}$ $d = \frac{11}{5}$

よって $A = \frac{1}{5}\begin{pmatrix} 14 & 2 \\ 2 & 11 \end{pmatrix}$

(2)

$|A - \lambda E| = \begin{vmatrix} \frac{14}{5} - \lambda & \frac{2}{5} \\ \frac{2}{5} & \frac{11}{5} - \lambda \end{vmatrix} = \left(\frac{14}{5} - \lambda\right)\left(\frac{11}{5} - \lambda\right) - \frac{4}{25} = \frac{154}{25} - 5\lambda + \lambda^2 - \frac{4}{25} = \lambda^2 - 5\lambda + 6$

$= (\lambda - 2)(\lambda - 3)$ $\lambda = 2, 3$

$\lambda = 3$ のとき

$\begin{pmatrix} -1 & 2 \\ 2 & -4 \end{pmatrix} \to \begin{pmatrix} 1 & -2 \\ 0 & 0 \end{pmatrix}$ $p_1 = C_1\begin{pmatrix} 2 \\ 1 \end{pmatrix}$

$\lambda = 2$ のとき

$\begin{pmatrix} 4 & 2 \\ 2 & 1 \end{pmatrix} \to \begin{pmatrix} 2 & 1 \\ 0 & 0 \end{pmatrix}$ $p_2 = C_2\begin{pmatrix} -1 \\ 2 \end{pmatrix}$

$P = \begin{pmatrix} 2 & -1 \\ 1 & 2 \end{pmatrix}$ $|P| = 4 + 1 = 5$ $P^{-1} = \frac{1}{5}\begin{pmatrix} 2 & 1 \\ -1 & 2 \end{pmatrix}$

$A^n = P\begin{pmatrix} 3^n & 0 \\ 0 & 2^n \end{pmatrix}P^{-1} = \begin{pmatrix} 2 & -1 \\ 1 & 2 \end{pmatrix}\begin{pmatrix} 3^n & 0 \\ 0 & 2^n \end{pmatrix}\frac{1}{5}\begin{pmatrix} 2 & 1 \\ -1 & 2 \end{pmatrix} = \frac{1}{5}\begin{pmatrix} 2 \cdot 3^n & -2^n \\ 3^n & 2^{n+1} \end{pmatrix}\begin{pmatrix} 2 & 1 \\ -1 & 2 \end{pmatrix}$

$= \frac{1}{5}\begin{pmatrix} 4 \cdot 3^n + 2^n & 2 \cdot 3^n - 2^{n+1} \\ 2 \cdot 3^n - 2^{n+1} & 3^n + 2^{n+2} \end{pmatrix}$

$P^n = A^n\begin{pmatrix} 5 \\ 5 \end{pmatrix} = \frac{1}{5}\begin{pmatrix} 4 \cdot 3^n + 2^n & 2 \cdot 3^n - 2^{n+1} \\ 2 \cdot 3^n - 2^{n+1} & 3^n + 2^{n+2} \end{pmatrix}\begin{pmatrix} 5 \\ 5 \end{pmatrix} = \begin{pmatrix} 6 \cdot 3^n - 2^n \\ 3 \cdot 3^n + 2 \cdot 2^n \end{pmatrix}$