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解説文ゼロで学ぶ行列の種類一覧

Last updated at Posted at 2019-03-14

正方行列

例1.

\begin{pmatrix}
  0 & 0 & 0 \\
  0 & 0 & 0 \\
  0 & 0 & 0 \\
\end{pmatrix}

例2.

\begin{pmatrix}
  1 & 2 & 3 & 4 \\
  4 & 3 & 2 & 1 \\
  4 & -4 & 4 & 4 \\
  1 & 1 & 1 & -1 \\
\end{pmatrix}

例3.

\begin{pmatrix}
  a_{11} & a_{12} & \cdots & a_{1n} \\
  a_{21} & a_{22} & \cdots & a_{2n} \\
  \vdots & \vdots & \ddots & \vdots \\
  a_{n1} & a_{n2} & \cdots & a_{nn} \\
\end{pmatrix}

Wikipedia - 正方行列

零行列

例1.

\begin{pmatrix}
  0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 \\
\end{pmatrix}

例2.

\begin{pmatrix}
  0 & 0 & 0 & \cdots & 0 \\
  0 & 0 & 0 & \cdots & 0 \\
  \vdots & \vdots & \vdots & \ddots & \vdots \\
  0 & 0 & 0 & \cdots & 0 \\
\end{pmatrix}

Wikipedia - 零行列

対角行列

例1.

\begin{pmatrix}
  1 & 0 & 0 \\
  0 & 2 & 0 \\
  0 & 0 & 3 \\
\end{pmatrix}

例2.

\begin{pmatrix}
  0 & 0 & 0 & 0 & 0 \\
  0 & 3 & 0 & 0 & 0 \\
  0 & 0 & -2 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 & -1 \\
\end{pmatrix}

例3.

\begin{pmatrix}
  c_{1} & 0 & 0 & 0 \\
  0 & c_{2} & 0 & 0 \\
  0 & 0 & \ddots & 0 \\
  0 & 0 & 0 & c_{n} \\
\end{pmatrix}

誤っている例1.

{\color{red}{

\begin{pmatrix}
  0 & 0 & 1 \\
  0 & 2 & 0 \\
  3 & 0 & 0 \\
\end{pmatrix}

}}

誤っている例2.

{\color{red}{

\begin{pmatrix}
  1 & 0 & 0 & 0 \\
  0 & 2 & 0 & 0 \\
  0 & 0 & 3 & 0 \\
\end{pmatrix}

}}

Wikipedia - 対角行列

三角行列

下三角行列の例1.

\begin{pmatrix}
  4 & 0 & 0 \\
  2 & 3 & 0 \\
  1 & 3 & 4 \\
\end{pmatrix}

下三角行列の例2.

\begin{pmatrix}
  a_{11} & 0 & 0 & 0 & 0 \\
  a_{21} & a_{22} & 0 & 0 & 0 \\
  a_{31} & a_{32} & \ddots & 0 & 0 \\
  \vdots & \vdots & \ddots & \ddots & 0 \\
  a_{n1} & a_{n2} & \dots & \cdots & a_{nn} \\
\end{pmatrix}

下三角行列の例3.

\begin{pmatrix}
  1 & 0 & 0 & 0 \\
  0 & 4 & 0 & 0 \\
  0 & 3 & 2 & 0 \\
  1 & 3 & 0 & 5 \\
\end{pmatrix}

上三角行列の例1.

\begin{pmatrix}
  a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\
  0 & a_{22} & a_{23} & \cdots & a_{2n} \\
  0 & 0 & \ddots & \ddots & \vdots \\
  0 & 0 & 0 & \ddots & \vdots \\
  0 & 0 & 0 & 0 & a_{nn} \\
\end{pmatrix}

Wikipedia - 三角行列

ハンケル行列

例1.

\begin{pmatrix}
  1 & 3 & 4 & 5 \\
  3 & 4 & 5 & 7 \\
  4 & 5 & 7 & 8 \\
  5 & 7 & 8 & 6 \\
\end{pmatrix}

例2.

\begin{pmatrix}
  a & b & c & d & e \\
  b & c & d & e & h \\
  c & d & e & h & g \\
  d & e & h & g & h \\
  e & h & g & h & i \\
\end{pmatrix}

誤った例1.

\color{red}{

\begin{pmatrix}
  e & d & c & b & a \\
  f & e & d & c & b \\
  g & f & e & d & c \\
  h & g & f & e & d \\
  i & h & g & f & e \\
\end{pmatrix}

}

Wikipedia - ハンケル行列

転置行列

例1.

A = \begin{pmatrix}
  1 & 2 & 3 \\
  4 & 5 & 6 \\
  7 & 8 & 9 \\
\end{pmatrix}

→

A^{\mathrm{T}} = \begin{pmatrix}
  1 & 4 & 7 \\
  2 & 5 & 8 \\
  3 & 6 & 9 \\
\end{pmatrix}

例2.

A = \begin{pmatrix}
  a & b & c \\
  d & e & f \\
\end{pmatrix}

→

A^{\mathrm{T}} = \begin{pmatrix}
  a & d \\
  b & e \\
  c & f \\
\end{pmatrix}

参考画像.

Wikipedia - 転置行列

複素共役行列

例1.

A = \begin{pmatrix}
  1 & -1 - \mathrm{i} & \\
  1 + \mathrm{i} & \mathrm{i} & \\
\end{pmatrix}

→

\overline{A} = \begin{pmatrix}
  1 & -1 + \mathrm{i} & \\
  1 - \mathrm{i} & -\mathrm{i} & \\
\end{pmatrix}

例2.

A = \begin{pmatrix}
  a_{11} + b_{11}\mathrm{i} & a_{12} + b_{12}\mathrm{i} \\
  a_{21} + b_{21}\mathrm{i} & a_{22} + b_{22}\mathrm{i} \\
  a_{31} + b_{31}\mathrm{i} & a_{32} + b_{32}\mathrm{i} \\
\end{pmatrix}

→

\overline{A} = \begin{pmatrix}
  a_{11} - b_{11}\mathrm{i} & a_{12} - b_{12}\mathrm{i} \\
  a_{21} - b_{21}\mathrm{i} & a_{22} - b_{22}\mathrm{i} \\
  a_{31} - b_{31}\mathrm{i} & a_{32} - b_{32}\mathrm{i} \\
\end{pmatrix}

(参考) Wikipedia - 複素共役

随伴行列

例1.

A = \begin{pmatrix}
  1 & -1 - \mathrm{i} & \\
  1 + \mathrm{i} & \mathrm{i} & \\
\end{pmatrix}

→

A^{\dagger} = \overline{A}^{\mathrm{T}} = \begin{pmatrix}
  1 & 1 - \mathrm{i} & \\
  -1 + \mathrm{i} & -\mathrm{i} & \\
\end{pmatrix}

例2.

A = \begin{pmatrix}
  a_{11} + b_{11}\mathrm{i} & \cdots & a_{1n} + b_{1n}\mathrm{i} & \\
  a_{21} + b_{21}\mathrm{i} & \cdots & a_{2n} + b_{2n}\mathrm{i} & \\
  a_{31} + b_{31}\mathrm{i} & \cdots & a_{3n} + b_{3n}\mathrm{i} & \\
\end{pmatrix}

→

A^{\dagger} = \overline{A}^{\mathrm{T}} = \begin{pmatrix}
  a_{11} - b_{11}\mathrm{i} & a_{21} - b_{21}\mathrm{i} & a_{31} - b_{31}\mathrm{i} \\
  \vdots & \vdots & \vdots \\
  a_{1n} - b_{1n}\mathrm{i} & a_{2n} - b_{2n}\mathrm{i} & a_{3n} - b_{3n}\mathrm{i} \\
\end{pmatrix}

Wikipedia - 随伴行列

対称行列

例1.

A = A^{\mathrm{T}}

例2.

A = \begin{pmatrix}
  1 & 7 & 3 \\
  7 & 4 & -5 \\
  3 & -5 & 6 \\
\end{pmatrix}

誤った例1.

\color{red}{

\begin{pmatrix}
  6 & 7 & 3 \\
  7 & 4 & -5 \\
  1 & -5 & 6 \\
\end{pmatrix}

}

Wikipedia - 対称行列

エルミート行列

例1.

A = A^{\mathrm{\dagger}}

例2.

A = \begin{pmatrix}
  3 & -3\mathrm{i} & 1 + \mathrm{i} & 2 - 4\mathrm{i} \\
  3\mathrm{i} & -3 & -5 + \mathrm{i} & 4\mathrm{i} \\
  1 - \mathrm{i} & -5 - \mathrm{i} & 0 & 1 + \mathrm{i} \\
  2 + 4\mathrm{i} & -4\mathrm{i} & 1 - \mathrm{i} & 4 \\
\end{pmatrix}

Wikipedia - エルミート行列

正規行列

例1.

A^{\mathrm{\dagger}}A = AA^{\mathrm{\dagger}}

例2.

A = \begin{pmatrix}
  1 & 2 \\
  2 & 1 \\
\end{pmatrix}

例3.

A = \begin{pmatrix}
  0 & 2 & 0 \\
  0 & 0 & 2 \\
  2 & 0 & 0 \\
\end{pmatrix}

Wikipedia - 正規行列
正規行列の性質

単位行列

例1.

\begin{pmatrix}
  1 & 0 & 0 & 0 & 0 \\
  0 & 1 & 0 & 0 & 0 \\
  0 & 0 & 1 & 0 & 0 \\
  0 & 0 & 0 & 1 & 0 \\
  0 & 0 & 0 & 0 & 1 \\
\end{pmatrix}

例2.

\begin{pmatrix}
  1 & 0 & \cdots & 0 \\
  0 & 1 & \cdots & 0 \\
  \vdots & \vdots & \ddots & \vdots \\
  0 & 0 & \cdots & 1 \\
\end{pmatrix}

誤った例1.

\color{red}{

\begin{pmatrix}
  0 & 0 & 0 & 1 \\
  0 & 0 & 1 & 0 \\
  0 & 1 & 0 & 0 \\
  1 & 0 & 0 & 0 \\
\end{pmatrix}

}

Wikipedia - 単位行列

正則行列、逆行列

例1.

AB = I = BA
  • $I$は単位行列
  • $A$は$B$の正則行列、$B$は$A$の逆行列
  • $B$は$A$の正則行列、$A$は$B$の逆行列

例2.

AA^{-1} = I = A^{-1}A
  • $A^{-1}$は$A$の逆行列

例3.

A = \begin{pmatrix}
  1 & 0 & 0 \\
  0 & 2 & 0 \\
  0 & 0 & -\frac{1}{5} \\
\end{pmatrix}

B = \begin{pmatrix}
  1 & 0 & 0 \\
  0 & \frac{1}{2} & 0 \\
  0 & 0 & -5 \\
\end{pmatrix}

\\

AB = \begin{pmatrix}
  1 & 0 & 0 \\
  0 & 1 & 0 \\
  0 & 0 & 1 \\
\end{pmatrix}

Wikipedia - 正則行列

直交行列

例1.

M^\mathrm{T}M = MM^\mathrm{T} = I

例2.

\begin{pmatrix}
  0 & 0 & 1 & 0 \\
  1 & 0 & 0 & 0 \\
  0 & 1 & 0 & 0 \\
  0 & 0 & 0 & 1 \\
\end{pmatrix}

WolframAplha - 上記の計算

例3.

\begin{pmatrix}
  \cos\theta & -\sin\theta \\
  \sin\theta & \cos\theta \\
\end{pmatrix}

Wikipedia - 直交行列

ユニタリ行列

例1.

U^\mathrm{\dagger}U = UU^\mathrm{\dagger} = I
  • $I$は単位行列
  • $U$がユニタリ行列

例2.

\begin{pmatrix}
  \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} \\
  \frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
\end{pmatrix}

WolframAplha - 上記の計算

Wikipedia - ユニタリ行列

射影行列

例1.

H = X(X^TX)^{–1}X^T

MathWorks - ハット行列とてこ比

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