行列の指数関数三角関数(正弦と余弦)、マクローリン展開、非零行列、3次、交代行列

学習環境

手を動かしてまなぶ線形代数 (藤岡敦(著)、裳華房)の第4章(行列の指数関数)、12(行列の指数関数)、チャレンジ問題、数物系の問12.6の解答を求めてみる。

\begin{array}{l} A = [\begin{matrix} 0 & a & b \\ - a & 0 & c \\ - b & - c & 0 \end{matrix}] \\ A^{2} = [\begin{matrix} 0 & a & b \\ - a & 0 & c \\ - b & - c & 0 \end{matrix}] [\begin{matrix} 0 & a & b \\ - a & 0 & c \\ - b & - c & 0 \end{matrix}] \\ = [\begin{matrix} - (a^{2} + b^{2}) & - b c & a c \\ - b c & - (a^{2} + c^{2}) & - a b \\ a c & - a b & - (b^{2} + c^{2}) \end{matrix}] \\ A^{3} = [\begin{matrix} - (a^{2} + b^{2}) & - b c & a c \\ - b c & - (a^{2} + c^{2}) & - a b \\ a c & - a b & - (b^{2} + c^{2}) \end{matrix}] [\begin{matrix} 0 & a & b \\ - a & 0 & c \\ - b & - c & 0 \end{matrix}] \\ = [\begin{matrix} 0 & - a (a^{2} + b^{2}) - a c^{2} & - b (a^{2} + b^{2}) - b c^{2} \\ a (a^{2} + c^{2}) + a b^{2} & 0 & - b^{2} c - c (a^{2} + c^{2}) \\ a^{2} b + b (b^{2} + c^{2}) & a^{2} c + c (b^{2} + c^{2}) & 0 \end{matrix}] \\ = [\begin{matrix} 0 & - a r^{2} & - b r^{2} \\ a r^{2} & 0 & - c r^{2} \\ b r^{2} & c r^{2} & 0 \end{matrix}] \\ = (- 1) r^{2} [\begin{matrix} 0 & a & b \\ - a & 0 & c \\ - b & - c & 0 \end{matrix}] \\ = (- 1) r^{2} A \\ A^{4} = (- 1) r^{2} A^{2} \end{array}

また、

\begin{array}{l} A^{2 n + 1} \\ = A^{2 (n - 1) + 1 + 2} \\ = A^{2 (n - 1) + 1} A^{2} \\ = {(- 1)}^{n - 1} r^{2 (n - 1)} A \cdot A^{2} \\ = {(- 1)}^{n - 1} r^{2 (n - 1)} A^{3} \\ = {(- 1)}^{n - 1} r^{2 (n - 1)} (- 1) r^{2} A \\ = {(- 1)}^{n} r^{2 n} A \\ A^{2 n + 2} \\ = A^{2 (n - 1) + 2 + 2} \\ = {(- 1)}^{n - 1} r^{2 (n - 1)} A^{2} A^{2} \\ = {(- 1)}^{n - 1} r^{2 (n - 1)} A^{4} \\ = {(- 1)}^{n - 1} r^{2 (n - 1)} (- 1) r^{2} A^{2} \\ = {(- 1)}^{n} r^{2 n} A^{2} \end{array}

よって、帰納法によりすべての正の自然数nに対して成り立つ。

ゆえに、

\begin{array}{l} \exp A \\ = E + \sum_{n = 1}^{\infty} \frac{1}{n!} A^{n} \\ = E + \sum_{n = 0}^{\infty} \frac{1}{(2 n + 1)!} A^{2 n + 1} + \sum_{n = 1}^{\infty} \frac{1}{(2 n)!} A^{2 n} \\ = E + \sum_{n = 0}^{\infty} \frac{1}{(2 n + 1)!} {(- 1)}^{n} r^{2 n} A + \sum_{n = 1}^{\infty} \frac{1}{(2 n)!} A^{2 (n - 1) + 2} \\ = E + \frac{1}{r} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n + 1)!} r^{2 n + 1} A + \sum_{n = 1}^{\infty} \frac{1}{(2 n)!} {(- 1)}^{(n - 1)} r^{2 (n - 1)} A^{2} \\ = E + \frac{\sin r}{r} A - \frac{1}{r^{2}} \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} r^{2 n} A^{2} \\ = E + \frac{\sin r}{r} A - \frac{1}{r^{2}} (- 1 + \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} r^{2 n}) A^{2} \\ = E + \frac{\sin r}{r} A + \frac{1 - \cos r}{r^{2}} A^{2} \end{array}