線形代数-1-6 〜正則の分割〜

どうも、いもけんぴぃです。

今回で第1章は最終節となります！

(1) $\begin{array}{r} (\begin{array}{cc} O & I \\ I & O \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} X & Y \\ Z & W \end{array}) \end{array}$ (2) $\begin{array}{r} (\begin{array}{cc} O & B \\ I & O \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} X & Y \\ Z & W \end{array}) \end{array}$

(3) $\begin{array}{r} (\begin{array}{cc} I & O \\ C & I \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} I & O \\ - C & I \end{array}) \end{array}$ (4) $\begin{array}{r} (\begin{array}{cc} A & O \\ O & I \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} I & O \\ C & I \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} I & O \\ O & D \end{array}) \end{array}$

<解説>
定義通りに計算していきます.

(1) $\begin{array}{r} (\begin{array}{cc} O & I \\ I & O \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} X & Y \\ Z & W \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} O \times X + I \times Z & O \times Y + I \times W \\ I \times X + O \times Z & I \times Y + O \times W \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} Z & W \\ X & Y \end{array}) \end{array}$

(2) $\begin{array}{r} (\begin{array}{cc} O & B \\ I & O \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} X & Y \\ Z & W \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} O \times X + B \times Z & O \times Y + B \times W \\ I \times X + O \times Z & I \times Y + O \times W \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} B Z & B W \\ X & Y \end{array}) \end{array}$

(3) $\begin{array}{r} (\begin{array}{cc} I & O \\ C & I \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} I & O \\ - C & I \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} I \times I + O \times (- C) & I \times O + O \times I \\ C \times I + I \times (- C) & C \times O + I \times I \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} I & O \\ O & I \end{array}) \end{array}$

(4) $\begin{array}{r} (\begin{array}{cc} A & O \\ O & I \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} I & O \\ C & I \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} I & O \\ O & D \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} A \times I + O \times C & A \times O + O \times I \\ O \times I + I \times C & O \times O + I \times I \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} I & O \\ O & D \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} A & O \\ C & I \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} I & O \\ O & D \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} A \times I + O \times O & A \times O + O \times D \\ C \times I + I \times O & C \times O + I \times D \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} A & O \\ C & D \end{array}) \end{array}$

問16

$m \times n$ 行列 $A = (a_{1}, a_{2}, \dots, a_{n})$ と $n$ 次列ベクトル $x = \begin{array}{r} (\begin{array}{cc} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{array}) \end{array}$ に対して,
$A x = x_{1} a_{1} + x_{2} a_{2} + \dots + x_{n} a_{n}$
となることを確かめよ.

<解説>
パッとみて「 $A$ が $m \times n$ 行列ってどういうこと!?」と思った方もいるかも知れませんが, $a_{i}$ は $A$ を列分割した $m \times 1$ 行列で,
$a_{i} = \begin{array}{r} (\begin{array}{cc} a_{1 i} \\ a_{2 i} \\ ⋮ \\ a_{m i} \end{array}) \end{array}$ のような行列です。

まず $A$ は $m \times n$ 行列, $x$ は $n \times 1$ 行列ですので積 $A x$ は定義されます。

$A x$ $=$ $\begin{array}{r} (\begin{array}{cc} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} \\ a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} \\ ⋮ \\ a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{m n} x_{n} \end{array}) \end{array}$

$=$ $x_{1} \begin{array}{r} (\begin{array}{cc} a_{11} \\ a_{21} \\ ⋮ \\ a_{m 1} \end{array}) \end{array}$ $+$ $x_{2} \begin{array}{r} (\begin{array}{cc} a_{12} \\ a_{22} \\ ⋮ \\ a_{m 2} \end{array}) \end{array}$ $+ \dots +$ $x_{n} \begin{array}{r} (\begin{array}{cc} a_{1 n} \\ a_{2 n} \\ ⋮ \\ a_{m n} \end{array}) \end{array}$