🔰matrix

LaTeX ⟩ matrix

\begin{matrix} a & b \\ c & d \end{matrix}      % matrix
\begin{bmatrix} a & b \\ c & d \end{bmatrix}    % [ ... ]
\begin{vmatrix} a & b \\ c & d \end{vmatrix}    % | ... |
\begin{pmatrix} a & b \\ c & d \end{pmatrix}    % ( ... )

example

code

$\begin{matrix} a & b \\ c & d \end{matrix}$

\begin{matrix} a & b \\ c & d \end{matrix}

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$

\begin{bmatrix} a & b \\ c & d \end{bmatrix}

$\begin{vmatrix} a & b \\ c & d \end{vmatrix}$

\begin{vmatrix} a & b \\ c & d \end{vmatrix}

$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$

% zero matrix
\begin{bmatrix}
  0 & 0 & 0 \\ 
  0 & 0 & 0 \\ 
  0 & 0 & 0
\end{bmatrix}

$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

% identity matrix
\begin{bmatrix}    
  1 & 0 & 0 \\    
  0 & 1 & 0 \\   
  0 & 0 & 1  
\end{bmatrix}

$\begin{bmatrix} a_{1} & & \\ & \ddots & \\ & & a_{n} \end{bmatrix}$

% diagonal matrix
\begin{bmatrix}
  a_{1} &      &     \\ 
  &     \ddots &     \\ 
  &     &      a_{n}
\end{bmatrix}

$\begin{Vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{Vmatrix}$

% V matrix
\begin{Vmatrix}
  a_{11} & a_{12} & a_{13} \\ 
  a_{21} & a_{22} & a_{23} \\
  a_{31} & a_{32} & a_{33} 
\end{Vmatrix}

$\begin{Bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{Bmatrix}$

% B matrix
\begin{Bmatrix}
  a_{11} & a_{12} & a_{13} \\ 
  a_{21} & a_{22} & a_{23} \\
  a_{31} & a_{32} & a_{33} 
\end{Bmatrix}

$\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$

👉 矩陣

% big matrix
\begin{bmatrix}
   a_{11} & a_{12} & \cdots & a_{1n} \\
   a_{21} & a_{22} & \cdots & a_{2n} \\
   \vdots & \vdots & \ddots & \vdots  \\
   a_{m1} & a_{m2} & \cdots & a_{mn} 
\end{bmatrix}

example

code

$\begin{bmatrix} x & y & z \end{bmatrix}$ $\begin{bmatrix} a_1 & \cdots & a_n \end{bmatrix}$

% row matrix
\begin{bmatrix} x & y & z \end{bmatrix}
\begin{bmatrix} a_1 & \cdots & a_n \end{bmatrix}

$\begin{bmatrix} x \\ y \\ z \end{bmatrix}$ , $\begin{bmatrix} a_1 \\ \vdots \\ a_n \end{bmatrix}$

% column matrix
\begin{bmatrix} x \\ y \\ z \end{bmatrix}
\begin{bmatrix} a_1 \\ \vdots \\ a_n \end{bmatrix}

$\begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}$

👉 內積

% dot product as matrix multiplication
\begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix}  
\begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}

$\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$

👉 空間外積

% cross product
\begin{vmatrix}    
  \mathbf{i} & \mathbf{j} & \mathbf{k} \\
  u_1 & u_2 & u_3 \\
  v_1 & v_2 & v_3 
\end{vmatrix}

$\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}$

% linear transformation
\begin{bmatrix}    
  a_1 & b_1 & c_1 \\
  a_2 & b_2 & c_2 \\
  a_3 & b_3 & c_3 
\end{bmatrix}
\begin{bmatrix} x \\ y \\ z \end{bmatrix}

👉 矩陣乘法表格化

$\mathbf{A}_{*\color{red}{k}} \mathbf{B}_{{\color{red}{k}}*}$

$\begin{pmatrix} & a_{1 {\color{red}{k}}}& \\ & \vdots & \\ \cdots & a_{i{\color{red}{k}}} & \cdots \\ & \vdots & \\ & a_{m{\color{red}{k}}} & \end{pmatrix} \begin{pmatrix} & & \vdots & & \\ b_{{\color{red}{k}}1} & \cdots & b_{{\color{red}{k}}j} & \cdots & b_{{\color{red}{k}}n}\\ & & \vdots & & \end{pmatrix}$

\begin{pmatrix}
 & a_{1 {\color{red}{k}}}& \\
 & \vdots  & \\
 \cdots  & a_{i{\color{red}{k}}} & \cdots \\
 & \vdots  & \\
 & a_{m{\color{red}{k}}} & 
\end{pmatrix}

\begin{pmatrix}
 &  & \vdots  &  & \\
 b_{{\color{red}{k}}1} & \cdots  & b_{{\color{red}{k}}j} & \cdots  & b_{{\color{red}{k}}n}\\
 &  & \vdots  &  & 
\end{pmatrix}

$\mathbf{AB} = \mathbf{A}_{*\color{red}{1}} \mathbf{B}_{{\color{red}{1}}*} + \mathbf{A}_{*\color{red}{2}} \mathbf{B}_{{\color{red}{2}}*} + \cdots + \mathbf{A}_{*\color{red}{p}} \mathbf{B}_{{\color{red}{p}}*}$

\mathbf{AB} =
  \mathbf{A}_{*\color{red}{1}} \mathbf{B}_{{\color{red}{1}}*}
+ \mathbf{A}_{*\color{red}{2}} \mathbf{B}_{{\color{red}{2}}*}
+ \cdots 
+ \mathbf{A}_{*\color{red}{p}} \mathbf{B}_{{\color{red}{p}}*}

👉 矩陣乘法

$\underbrace{\begin{pmatrix} & & \vdots & \\ a_{i1} & a_{i2} & \cdots & a_{ip} \\ & & \vdots & \end{pmatrix}}_{m\times p\ \text{matrix}} \underbrace{\begin{pmatrix} & b_{1j} & \\ & b_{2j} & \\ \cdots & \vdots & \cdots \\ & b_{pj} & \end{pmatrix}}_{p\times n\ \text{matrix}} = \underbrace{\begin{pmatrix} & \vdots & \\ \cdots & c_{ij} & \cdots \\ & \vdots & \end{pmatrix}}_{m\times n\ \text{matrix}}$

\underbrace{\begin{pmatrix}
        &        & \vdots  &        \\
 a_{i1} & a_{i2} & \cdots  & a_{ip} \\
        &        & \vdots  & 
\end{pmatrix}}_{m\times p\ \text{matrix}}
\underbrace{\begin{pmatrix}
         & b_{1j} &        \\
         & b_{2j} &        \\
 \cdots  & \vdots & \cdots \\
         & b_{pj} & 
\end{pmatrix}}_{p\times n\ \text{matrix}}
=
\underbrace{\begin{pmatrix}
        & \vdots &        \\
 \cdots & c_{ij} & \cdots \\
        & \vdots & 
\end{pmatrix}}_{m\times n\ \text{matrix}}

$\mathbf{A}_{{\color{red}{i}}*} \mathbf{B}_{*\color{red}{j}}$ 👉 行向量

$\begin{pmatrix} & & \vdots & \\ a_{i1} & a_{i2} & \cdots & a_{ip}\\ & & \vdots & \end{pmatrix}\begin{pmatrix} & b_{1j} & \\ & b_{2j} & \\ \cdots & \vdots & \cdots \\ & b_{pj} & \end{pmatrix}$

\begin{pmatrix}
   &  & \vdots  & \\
   a_{i1} & a_{i2} & \cdots  & a_{ip}\\
   &  & \vdots  & 
\end{pmatrix}
\begin{pmatrix}
   & b_{1j} & \\
   & b_{2j} & \\
\cdots  & \vdots  & \cdots \\
   & b_{pj} & 
\end{pmatrix}

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Last updated 2 years ago

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