線形代数-1-基礎問題-3

どうも、いもけんぴぃです。

更新が非常に遅くなってしまいすみません...
2次複素行列が3次複素行列だったらマジで一生手をつけてなかったです

今回はひたすら計算です

目次

• 問1.4
• 次回予告

問1.4

<解説>

$(1)$$\sigma^2_{x}$$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0\times0 + 1\times1 & 0\times1 + 1\times0 \\ 1\times0 + 0\times1 & 1\times1 + 0\times0 \\ \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right) \end{eqnarray} $

$=$$I$

$\sigma^2_{y}$$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0\times0 + -i\times i & 0\times(-i) + (-i)\times0 \\ i\times0 + 0\times i & i\times (-i) + 0\times0 \\ \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right) \end{eqnarray} $

$=$$I$

$\sigma^2_{z}$$=$$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 1\times1 + 0\times0 & 1\times0 + 0\times(-1) \\ 0\times1 + -1\times0 & 0\times0 + (-1)\times(-1) \\ \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right) \end{eqnarray} $

$=$$I$

$(2)$
$\sigma_{x}\sigma_{y} + \sigma_{y}\sigma_{x}$$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right) \end{eqnarray} $$+$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0\times0 + 1\times i & 0\times(-i) + 1\times 0 \\ 1\times0 + 0\times i & 1\times(-i) + 0\times 0 \end{array} \right) \end{eqnarray} $$+$$ \begin{eqnarray} \left( \begin{array}{cc} 0\times0 + (-i)\times 1 & 0\times1 + (-i)\times 0 \\ i\times0 + 0\times1 & i\times1 + 0\times 0 \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right) \end{eqnarray} $$+$$ \begin{eqnarray} \left( \begin{array}{cc} -i & 0 \\ 0 & i \end{array} \right) \end{eqnarray} $

$=$$O$
$\sigma_{y}\sigma_{z} + \sigma_{z}\sigma_{y}$$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \end{eqnarray} $$+$$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0\times1 + (-i)\times 0 & 0\times0 + (-i)\times (-1) \\ i\times1 + 0\times 0 & i\times0 + 0\times (-1) \end{array} \right) \end{eqnarray} $$+$$ \begin{eqnarray} \left( \begin{array}{cc} 1\times0 + 0\times i & 1\times(-i) + 0\times 0 \\ 0\times0 + (-1)\times i & 0\times(-i) + (-1)\times 0 \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & i \\ i & 0 \end{array} \right) \end{eqnarray} $$+$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ -i & 0 \end{array} \right) \end{eqnarray} $

$=$$O$
$\sigma_{z}\sigma_{x} + \sigma_{x}\sigma_{z}$$=$$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right) \end{eqnarray} $$+$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 1\times0 + 0\times1 & 1\times1 + 0\times 0 \\ 0\times0 + (-1)\times1 & 0\times1 + (-1)\times 0 \end{array} \right) \end{eqnarray} $$+$$ \begin{eqnarray} \left( \begin{array}{cc} 0\times1 + 1\times0 & 0\times0 + 1\times(-1) \\ 1\times1 + 0\times0 & 1\times0 + 0\times(-1) \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) \end{eqnarray} $$+$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) \end{eqnarray} $

$=$$O$

$(3)$
$\sigma_{x}\sigma_{y} - \sigma_{y}\sigma_{x}$$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right) \end{eqnarray} $$-$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0\times0 + 1\times i & 0\times(-i) + 1\times 0 \\ 1\times0 + 0\times i & 1\times(-i) + 0\times 0 \end{array} \right) \end{eqnarray} $$-$$ \begin{eqnarray} \left( \begin{array}{cc} 0\times0 + (-i)\times 1 & 0\times1 + (-i)\times 0 \\ i\times0 + 0\times1 & i\times1 + 0\times 0 \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right) \end{eqnarray} $$-$$ \begin{eqnarray} \left( \begin{array}{cc} -i & 0 \\ 0 & i \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 2i & 0 \\ 0 & -2i \end{array} \right) \end{eqnarray} $

$=$$2i$$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \end{eqnarray} $

$=$$2i\sigma_{z}$

$\sigma_{y}\sigma_{z} - \sigma_{z}\sigma_{y}$$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \end{eqnarray} $$-$$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ i & 0 \\ \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0\times1 + (-i)\times 0 & 0\times0 + (-i)\times (-1) \\ i\times1 + 0\times 0 & i\times0 + 0\times (-1) \end{array} \right) \end{eqnarray} $$-$$ \begin{eqnarray} \left( \begin{array}{cc} 1\times0 + 0\times i & 1\times(-i) + 0\times 0 \\ 0\times0 + (-1)\times i & 0\times(-i) + (-1)\times 0 \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & i \\ i & 0 \end{array} \right) \end{eqnarray} $$-$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ -i & 0 \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 2i \\ 2i & 0 \end{array} \right) \end{eqnarray} $

$=$$2i$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \end{eqnarray} $

$=$$2i\sigma_{x}$

$\sigma_{z}\sigma_{x} - \sigma_{x}\sigma_{z}$$=$$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right) \end{eqnarray} $$-$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right) \end{eqnarray} $$ \begin{eqnarray} \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 1\times0 + 0\times1 & 1\times1 + 0\times 0 \\ 0\times0 + (-1)\times1 & 0\times1 + (-1)\times 0 \end{array} \right) \end{eqnarray} $$-$$ \begin{eqnarray} \left( \begin{array}{cc} 0\times1 + 1\times0 & 0\times0 + 1\times(-1) \\ 1\times1 + 0\times0 & 1\times0 + 0\times(-1) \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) \end{eqnarray} $$-$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) \end{eqnarray} $

$=$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & 2 \\ -2 & 0 \end{array} \right) \end{eqnarray} $

$=$$2i$$ \begin{eqnarray} \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right) \end{eqnarray} $

$=$$2i\sigma_{y}$

次回予告

次回は問1.5,問1.6,問1.7の3本立てでお送りします。
次回もまた見てくださいね！
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