第17章 17.29

(1)

$J^2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = -E$

(2)

$P(x) = (X - aE)^2 + (bE)^2 = X^2 - 2aX + a^2E + b^2E$

これが $X^2 - 4X + 13E$ となるので

$2a = 4$ → $a = 2$

$a^2E + b^2E = 4E + b^2E = 13E$ → $b^2 = 9$ → $b = \pm 3$

(3)

$X = aE + bJ = 2 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \pm 3 \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \pm \begin{pmatrix} 0 & -3 \\ 3 & 0 \end{pmatrix} = \begin{pmatrix} 2 & \mp 3 \\ \pm 3 & 2 \end{pmatrix}$

$P(x) = (X - aE)^2 + (bE)^2 = (aE + bJ - aE)^2 + (bE)^2 = b^2J^2 + b^2E = -b^2E + b^2E = 0$

(4)

$|X - \lambda E| = \begin{vmatrix} 2 - \lambda & \mp 3 \\ \pm 3 & 2 - \lambda \end{vmatrix} = (2 - \lambda)^2 + 9 = \lambda^2 - 4\lambda + 13 = 0 \quad \lambda = 2 \pm \sqrt{4 - 13} = 2 \pm 3i$

(5)

$V^2 = \begin{pmatrix} 3 & -2 \\ 2 & 3 \end{pmatrix} \begin{pmatrix} 3 & -2 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 5 & -12 \\ 12 & 5 \end{pmatrix}$

$Q(V) = V^2 + cV + dE = \begin{pmatrix} 5 & -12 \\ 12 & 5 \end{pmatrix} + c \begin{pmatrix} 3 & -2 \\ 2 & 3 \end{pmatrix} + d \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 5 + 3c + d & -12 - 2c \\ 12 + 2c & 5 + 3c + d \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

$12 + 2c = 0$ → $c = -6$

$5 + 3c + d = 1$ → $5 - 18 + d = 1$ → $d = 14$