講義（山本裕樹）

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====== 行列 ======

===== 行列の種類 =====

==== 零行列 ====

\[
\pmb{\textit{O}}
=\left(
\begin{array}{ccccc}
0 & 0 & \cdots & 0 & 0\\
0 & 0 & \cdots & 0 & 0\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
0 & 0 & \cdots & 0 & 0\\
0 & 0 & \cdots & 0 & 0
\end{array}
\right)
\]

----
==== 単位行列 ====

\[
\pmb{\textit{I}}
=\left(
\begin{array}{ccccc}
1 & 0 & \cdots & 0 & 0\\
0 & 1 & \cdots & 0 & 0\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
0 & 0 & \cdots & 1 & 0\\
0 & 0 & \cdots & 0 & 1
\end{array}
\right)
\]

===== 行列の演算 =====
次の行列を考える。

\[
\pmb{\textit{A}}=\left(
\begin{array}{cc}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{array}
\right)
,\quad
\pmb{\textit{B}}=\left(
\begin{array}{cc}
b_{11} & b_{12}\\
b_{21} & b_{22}
\end{array}
\right)
\]

\[
\pmb{\textit{x}}=\left(
\begin{array}{c}
x_{1}\\
x_{2}
\end{array}
\right)
,\quad
\pmb{\textit{y}}=\left(
\begin{array}{c}
y_{1}\\
y_{2}
\end{array}
\right)
\]

----
==== 転置 ====

\[
\pmb{\textit{A}}^{T}=\left(
\begin{array}{cc}
a_{11} & a_{21}\\
a_{12} & a_{22}
\end{array}
\right)
\]

\[
\pmb{\textit{x}}^{T}=\left(
\begin{array}{cc}
x_{1} & x_{2}
\end{array}
\right)
\]

----
==== 加法 ====
\[
\pmb{\textit{A}}+\pmb{\textit{B}}
=\left(
\begin{array}{cc}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{array}
\right)
+\left(
\begin{array}{cc}
b_{11} & b_{12}\\
b_{21} & b_{22}
\end{array}
\right)
=\left(
\begin{array}{cc}
a_{11}+b_{11} & a_{12}+b_{12}\\
a_{21}+b_{21} & a_{22}+b_{22}
\end{array}
\right)
\]

\[
\pmb{\textit{x}}+\pmb{\textit{y}}
=\left(
\begin{array}{c}
x_{1}\\
x_{2}
\end{array}
\right)
+\left(
\begin{array}{c}
y_{1}\\
y_{2}
\end{array}
\right)
=\left(
\begin{array}{c}
x_{1}+y_{1}\\
x_{2}+y_{2}
\end{array}
\right)
\]
----
==== スカラー倍 ====

$\lambda$ はスカラー（行列ではない）の定数とする。

\[
\lambda\pmb{\textit{A}}
=\lambda\left(
\begin{array}{cc}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{array}
\right)
=\left(
\begin{array}{cc}
\lambda a_{11} & \lambda a_{12}\\
\lambda a_{21} & \lambda a_{22}
\end{array}
\right)
\]

\[
\lambda\pmb{\textit{x}}
=\lambda\left(
\begin{array}{cc}
x_{1}\\
x_{2}
\end{array}
\right)
=\left(
\begin{array}{cc}
\lambda x_{1}\\
\lambda x_{2}
\end{array}
\right)
\]



----
==== 積 ====

\[
\pmb{\textit{A}}\pmb{\textit{B}}
=\left(
\begin{array}{cc}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{array}
\right)
\left(
\begin{array}{cc}
b_{11} & b_{12}\\
b_{21} & b_{22}
\end{array}
\right)
=\left(
\begin{array}{cc}
a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22}\\
a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22}
\end{array}
\right)
\]

\[
\pmb{\textit{B}}\pmb{\textit{A}}
=\left(
\begin{array}{cc}
b_{11} & b_{12}\\
b_{21} & b_{22}
\end{array}
\right)
\left(
\begin{array}{cc}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{array}
\right)
=\left(
\begin{array}{cc}
b_{11}a_{11}+b_{12}a_{21} & b_{11}a_{12}+b_{12}a_{22}\\
b_{21}a_{11}+b_{22}a_{21} & b_{21}a_{12}+b_{22}a_{22}
\end{array}
\right)
\]

\[
\pmb{\textit{A}}\pmb{\textit{x}}
=\left(
\begin{array}{cc}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{array}
\right)
\left(
\begin{array}{c}
x_{1}\\
x_{2}
\end{array}
\right)
=\left(
\begin{array}{c}
a_{11}x_{1}+a_{12}x_{2}\\
a_{21}x_{1}+a_{22}x_{2}
\end{array}
\right)
\]

\[
\pmb{\textit{x}}^{T}\pmb{\textit{A}}
=\left(
\begin{array}{cc}
x_{1} & x_{2}
\end{array}
\right)
\left(
\begin{array}{cc}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{array}
\right)
=\left(
\begin{array}{cc}
a_{11}x_{1}+a_{21}x_{2} & a_{12}x_{1}+a_{22}x_{2}
\end{array}
\right)
\]

\[
\pmb{\textit{x}}^{T}\pmb{\textit{y}}
=\left(
\begin{array}{cc}
x_{1} & x_{2}
\end{array}
\right)
\left(
\begin{array}{c}
y_{1}\\
y_{2}
\end{array}
\right)
=x_{1}y_{1}+x_{2}y_{2}
\]

===== 逆行列 =====

次の行列を考える。
\[
\pmb{\textit{A}}=\left(
\begin{array}{cc}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{array}
\right)
\]

\[
\pmb{\textit{A}}^{-1}
=
\frac{1}{a_{11}a_{22}-a_{12}a_{21}}
\left(
\begin{array}{cc}
a_{22} & -a_{12}\\
-a_{21} & a_{11}
\end{array}
\right)
\]

$\pmb{\textit{A}}^{-1}$ は $\pmb{\textit{A}}$ の逆行列という。
ただし、$a_{11}a_{22}-a_{12}a_{21}=0$ のときは逆行列は存在しない。

\[
\pmb{\textit{A}}^{-1}\pmb{\textit{A}}
=
\frac{1}{a_{11}a_{22}-a_{12}a_{21}}
\left(
\begin{array}{cc}
a_{22} & -a_{12}\\
-a_{21} & a_{11}
\end{array}
\right)
\left(
\begin{array}{cc}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{array}
\right)
=\left(
\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}
\right)
=\pmb{\textit{I}}
\]

\[
\pmb{\textit{A}}\pmb{\textit{A}}^{-1}
=
\pmb{\textit{I}}
\]