有没有办法自动转置用 Latex 编写的矩阵？

Question 1

这是一个实现expl3：

\documentclass{article}
\usepackage{xparse}
\usepackage{amsmath}
\usepackage{environ}

\ExplSyntaxOn
\NewEnviron{bmatrixT}
 {
  \marine_transpose:V \BODY
 }

\int_new:N \l_marine_transpose_row_int
\int_new:N \l_marine_transpose_col_int
\seq_new:N \l_marine_transpose_rows_seq
\seq_new:N \l_marine_transpose_arow_seq
\prop_new:N \l_marine_transpose_matrix_prop
\tl_new:N \l_marine_transpose_last_tl
\tl_new:N \l_marine_transpose_body_tl

\cs_new_protected:Nn \marine_transpose:n
 {
  \seq_set_split:Nnn \l_marine_transpose_rows_seq { \\ } { #1 }
  \int_zero:N \l_marine_transpose_row_int
  \prop_clear:N \l_marine_transpose_matrix_prop
  \seq_map_inline:Nn \l_marine_transpose_rows_seq
   {
    \int_incr:N \l_marine_transpose_row_int
    \int_zero:N \l_marine_transpose_col_int
    \seq_set_split:Nnn \l_marine_transpose_arow_seq { & } { ##1 }
    \seq_map_inline:Nn \l_marine_transpose_arow_seq
     {
      \int_incr:N \l_marine_transpose_col_int
      \prop_put:Nxn \l_marine_transpose_matrix_prop
       {
        \int_to_arabic:n { \l_marine_transpose_row_int }
        ,
        \int_to_arabic:n { \l_marine_transpose_col_int }
       }
       { ####1 }
     }
   }
   \tl_clear:N \l_marine_transpose_body_tl
   \int_step_inline:nnnn { 1 } { 1 } { \l_marine_transpose_col_int }
    {
     \int_step_inline:nnnn { 1 } { 1 } { \l_marine_transpose_row_int }
      {
       \tl_put_right:Nx \l_marine_transpose_body_tl
        {
         \prop_item:Nn \l_marine_transpose_matrix_prop { ####1,##1 }
         \int_compare:nF { ####1 = \l_marine_transpose_row_int } { & }
        }
      }
     \tl_put_right:Nn \l_marine_transpose_body_tl { \\ }
    }
   \begin{bmatrix}
   \l_marine_transpose_body_tl
   \end{bmatrix}
 }
\cs_generate_variant:Nn \marine_transpose:n { V }
\cs_generate_variant:Nn \prop_put:Nnn { Nx }
\ExplSyntaxOff

\begin{document}
\[
\begin{bmatrix}
   0 & -1 & -1 \\
   -1 & -1 & 0 \\
   -1 & 0 & 1 \\
   1 & 0 & 0 \\
   0 & 0 & -1 \\
   -1 & 2 & 1 
\end{bmatrix}^T
=
\begin{bmatrixT}
   0 & -1 & -1 \\
   -1 & -1 & 0 \\
   -1 & 0 & 1 \\
   1 & 0 & 0 \\
   0 & 0 & -1 \\
   -1 & 2 & 1 
\end{bmatrixT}
\]

\end{document}

Answer

这是一个实现expl3：

\documentclass{article}
\usepackage{xparse}
\usepackage{amsmath}
\usepackage{environ}

\ExplSyntaxOn
\NewEnviron{bmatrixT}
 {
  \marine_transpose:V \BODY
 }

\int_new:N \l_marine_transpose_row_int
\int_new:N \l_marine_transpose_col_int
\seq_new:N \l_marine_transpose_rows_seq
\seq_new:N \l_marine_transpose_arow_seq
\prop_new:N \l_marine_transpose_matrix_prop
\tl_new:N \l_marine_transpose_last_tl
\tl_new:N \l_marine_transpose_body_tl

\cs_new_protected:Nn \marine_transpose:n
 {
  \seq_set_split:Nnn \l_marine_transpose_rows_seq { \\ } { #1 }
  \int_zero:N \l_marine_transpose_row_int
  \prop_clear:N \l_marine_transpose_matrix_prop
  \seq_map_inline:Nn \l_marine_transpose_rows_seq
   {
    \int_incr:N \l_marine_transpose_row_int
    \int_zero:N \l_marine_transpose_col_int
    \seq_set_split:Nnn \l_marine_transpose_arow_seq { & } { ##1 }
    \seq_map_inline:Nn \l_marine_transpose_arow_seq
     {
      \int_incr:N \l_marine_transpose_col_int
      \prop_put:Nxn \l_marine_transpose_matrix_prop
       {
        \int_to_arabic:n { \l_marine_transpose_row_int }
        ,
        \int_to_arabic:n { \l_marine_transpose_col_int }
       }
       { ####1 }
     }
   }
   \tl_clear:N \l_marine_transpose_body_tl
   \int_step_inline:nnnn { 1 } { 1 } { \l_marine_transpose_col_int }
    {
     \int_step_inline:nnnn { 1 } { 1 } { \l_marine_transpose_row_int }
      {
       \tl_put_right:Nx \l_marine_transpose_body_tl
        {
         \prop_item:Nn \l_marine_transpose_matrix_prop { ####1,##1 }
         \int_compare:nF { ####1 = \l_marine_transpose_row_int } { & }
        }
      }
     \tl_put_right:Nn \l_marine_transpose_body_tl { \\ }
    }
   \begin{bmatrix}
   \l_marine_transpose_body_tl
   \end{bmatrix}
 }
\cs_generate_variant:Nn \marine_transpose:n { V }
\cs_generate_variant:Nn \prop_put:Nnn { Nx }
\ExplSyntaxOff

\begin{document}
\[
\begin{bmatrix}
   0 & -1 & -1 \\
   -1 & -1 & 0 \\
   -1 & 0 & 1 \\
   1 & 0 & 0 \\
   0 & 0 & -1 \\
   -1 & 2 & 1 
\end{bmatrix}^T
=
\begin{bmatrixT}
   0 & -1 & -1 \\
   -1 & -1 & 0 \\
   -1 & 0 & 1 \\
   1 & 0 & 0 \\
   0 & 0 & -1 \\
   -1 & 2 & 1 
\end{bmatrixT}
\]

\end{document}

Question 2

如果你将数学和 LaTeX 混合使用，那么你应该熟悉该sagetex软件包。CTAN 链接是这里。这为您在使用 LaTeX 时提供了计算机代数系统的强大功能。

\documentclass{article}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{sagetex}
\begin{document}
\begin{sagesilent}
latex.matrix_delimiters(left='[', right=']')
A = Matrix([[0,-1,-1],[-1,-1,0],[-1,0,1],[1,0,0],[0,0,-1],[-1,2,1]])
\end{sagesilent}
If $A = \sage{A}$ and $A^{\intercal} = \sage{A.transpose()}$ then 
$A \cdot A^{\intercal} = \sage{A*A.transpose()}$ and 
$A^{\intercal}\cdot A = \sage{A.transpose()*A}$
\end{document}

数学定义和计算通常在sagesilent模式下进行。当您需要计算时，请使用 \sage{} 来获取它。

代码输出如下所示：

如您所见，根据您输入的矩阵 A，Sage 负责确定转置和矩阵乘法，从而最大限度地减少错误。LaTex 和 Sage 都有文档记录这里。可以找到一些有关矩阵的 Sage 文档这里和这里。

唯一的缺点是你需要访问 Sage。你可以下载并安装在你的电脑上，但让它工作并不总是那么简单。但如果你不介意在云端工作，那么你可以获得一个免费的 Sagemath Cloud 帐户这里。这真的很简单，您只需几分钟就可以启动并运行。

Answer

如果你将数学和 LaTeX 混合使用，那么你应该熟悉该sagetex软件包。CTAN 链接是这里。这为您在使用 LaTeX 时提供了计算机代数系统的强大功能。

\documentclass{article}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{sagetex}
\begin{document}
\begin{sagesilent}
latex.matrix_delimiters(left='[', right=']')
A = Matrix([[0,-1,-1],[-1,-1,0],[-1,0,1],[1,0,0],[0,0,-1],[-1,2,1]])
\end{sagesilent}
If $A = \sage{A}$ and $A^{\intercal} = \sage{A.transpose()}$ then 
$A \cdot A^{\intercal} = \sage{A*A.transpose()}$ and 
$A^{\intercal}\cdot A = \sage{A.transpose()*A}$
\end{document}

数学定义和计算通常在sagesilent模式下进行。当您需要计算时，请使用 \sage{} 来获取它。

代码输出如下所示：

如您所见，根据您输入的矩阵 A，Sage 负责确定转置和矩阵乘法，从而最大限度地减少错误。LaTex 和 Sage 都有文档记录这里。可以找到一些有关矩阵的 Sage 文档这里和这里。

唯一的缺点是你需要访问 Sage。你可以下载并安装在你的电脑上，但让它工作并不总是那么简单。但如果你不介意在云端工作，那么你可以获得一个免费的 Sagemath Cloud 帐户这里。这真的很简单，您只需几分钟就可以启动并运行。

Question 3

更新：更好的注释代码，带有\Mar@前缀的更安全的内部宏名称，添加表格，“宏”和“文件”示例。

这是一个基于可扩展宏的方法\Transpose。可以在pmatrix、、bmatrix... 或tabular类似环境中使用它。

该宏的用途是，\Transpose {rows separated by \\}每行都用分隔符书写&，最后一行应该有不 \\。

扩展\Transpose分为两个步骤。有些项目可以移除括号，我并没有深入讨论细节，无论如何，表格单元格具有局部范围。

但我认为使用文本编辑器宏会更好；对于 e-Lisp 知识渊博的人来说，在 Emacs 缓冲区中肯定可以做到这一点。事实上，我猜典型的用例是使用大型表格材料，需要在转置后进一步编辑，然后重新插入 TeX 源。

作为学习 e-Lisp 的替代方法，人们可以使用\Transpose它在草稿运行中将其输出写入某个辅助文件，以便用户可以恢复转置的数据。这也在下一个代码示例中进行了说明。

\documentclass{article}

\usepackage{amsmath}
\usepackage{array}

\makeatletter
\catcode`! 3
\catcode0 12

% The update chooses safer macro names.

% It could be useful to allow #1 to be itself a macro expanding
% to some rows. One could replace \Mar@DoOneRow by 
% \expandafter\Mar@DoOneRow but then general usage must be aware
% of this one-expansion of contents.

% Usage: \Transpose { &-separated cells \\ &-separated cells \\ ...
%                     ...\\&-separated cells and *no* final \\ }

% Can be used inside amsmath matrices as well as inside LaTeX tabular's
% 
\def\Transpose #1{\romannumeral0\expandafter
                  \Mar@Transpose@a\romannumeral`^^@\Mar@DoOneRow #1\\!\\}

\def\Mar@DoOneRow #1\\{\Mar@DoOneRow@a {}#1&^^@&}%

\def\Mar@DoOneRow@a #1#2&{%
    \if^^@\detokenize{#2}\expandafter\@gobble\fi
    \Mar@DoOneRow@a {#1#2\\}%
}%

% unbraced #1 here will end in \\^^@\\
\def\Mar@Transpose@a #1#2\\{\ifx!#2\expandafter\Mar@FinishTranspose\fi
    \expandafter\Mar@Transpose@b\romannumeral`^^@\Mar@DoOneRow@a {}#2&^^@&#1}

% unbraced #1 here will end with \\^^@\\
% It represents a new transposed row with an extra ^^@ cell.
% The #2 here ends in \\. It holds the first already transposed rows.
\def\Mar@Transpose@b #1#2^^@\\{\Mar@Join {}#2^^@!#1}

% when #3=^^@ happens after #4\\ there is still ^^@\\
% update slightly simplifies \Mar@Join.
\def\Mar@Join #1#2\\#3!#4\\%
   {\if^^@\detokenize{#3}\expandafter\Mar@EndJoin\fi
    \Mar@Join {#1#2&#4\\}#3!}%

% unbraced #1 ends with \\
\def\Mar@EndJoin\Mar@Join #1^^@!^^@\\{\Mar@Transpose@a {#1^^@\\}}

\def\Mar@FinishTranspose
    #1&^^@&#2\\^^@\\{ #2}% the \\^^@\\ pattern helps removing a final \\

\catcode`! 12
\catcode0 15 % null character "invalid": this is default LaTeX setting.
\makeatother

\begin{document}

A bmatrix example:
\[
\begin{bmatrix}
    0 & -1 & -1 \\
   -1 & -1 & 0  \\
   -1 & 0  & 1  \\
    1 & 0  & 0  \\
    0 & 0  & -1 \\
   -1 & 2  & 1
\end{bmatrix}\longrightarrow
\begin{bmatrix}
  \Transpose {
    0 & -1 & -1 \\
   -1 & -1 & 0  \\
   -1 & 0  & 1  \\
    1 & 0  & 0  \\
    0 & 0  & -1 \\
   -1 & 2  & 1 }
\end{bmatrix}
\]

% works the same.
% \[
% \begin{pmatrix}
%   \Transpose {
%     0 & -1 & -1 \\
%    -1 & -1 & 0  \\
%    -1 & 0  & 1  \\
%     1 & 0  & 0  \\
%     0 & 0  & -1 \\
%    -1 & 2  & 1 }
% \end{pmatrix}
% \]

A tabular example:

\begin{center}
\begin{tabular}{c|>{\bfseries}c|>{\itshape}c}
  L1 & L2 & L3 \\
  M1 & M2 & M3 \\
  N1 & N2 & N3
\end{tabular}
${}\longrightarrow{}$
\begin{tabular}{c|>{\bfseries}c|>{\itshape}c}
  \Transpose{
  L1 & L2 & L3 \\
  M1 & M2 & M3 \\
  N1 & N2 & N3
}% <- this to avoid a spurious space which tabular does not remove
\end{tabular}
\end{center}

A macro-level example to illustrate that two expansion steps are enough and
that contents are not prematurely expanded:

% spaces from endlines and around tabs
% which disappear in tabular/matrix treatments
% are not trimmed by \Transpose itself 

\def\www {  
  \Arbitrary L1 & \Stuff L2 & \Here L3 \\
  M1 & M2 & M3 \\
  N1 & N2 & N3
}

\expandafter\expandafter\expandafter\def
\expandafter\expandafter\expandafter\zzz
\expandafter\expandafter\expandafter{\Transpose {
  \Arbitrary L1 & \Stuff L2 & \Here L3 \\
  M1 & M2 & M3 \\
  N1 & N2 & N3
}}

\ttfamily

\meaning\www

becomes

\meaning\zzz

% And finally a "write to file" example
\newwrite\testout

\immediate\openout\testout=\jobname-tranpose.out

\immediate\write\testout{\unexpanded\expandafter\expandafter\expandafter
   {\Transpose  {
  \Arbitrary L1 & \Stuff L2 & \Here L3 \\
  M1 & M2 & M3 \\
  N1 & N2 & N3
}}}

\end{document}

“filename-transpose.out”文件包含：

 \Arbitrary L1 & M1 & N1 \\ \Stuff L2 & M2 & N2 \\ \Here L3 & M3 & N3

Answer

更新：更好的注释代码，带有\Mar@前缀的更安全的内部宏名称，添加表格，“宏”和“文件”示例。

这是一个基于可扩展宏的方法\Transpose。可以在pmatrix、、bmatrix... 或tabular类似环境中使用它。

该宏的用途是，\Transpose {rows separated by \\}每行都用分隔符书写&，最后一行应该有不 \\。

扩展\Transpose分为两个步骤。有些项目可以移除括号，我并没有深入讨论细节，无论如何，表格单元格具有局部范围。

但我认为使用文本编辑器宏会更好；对于 e-Lisp 知识渊博的人来说，在 Emacs 缓冲区中肯定可以做到这一点。事实上，我猜典型的用例是使用大型表格材料，需要在转置后进一步编辑，然后重新插入 TeX 源。

作为学习 e-Lisp 的替代方法，人们可以使用\Transpose它在草稿运行中将其输出写入某个辅助文件，以便用户可以恢复转置的数据。这也在下一个代码示例中进行了说明。

\documentclass{article}

\usepackage{amsmath}
\usepackage{array}

\makeatletter
\catcode`! 3
\catcode0 12

% The update chooses safer macro names.

% It could be useful to allow #1 to be itself a macro expanding
% to some rows. One could replace \Mar@DoOneRow by 
% \expandafter\Mar@DoOneRow but then general usage must be aware
% of this one-expansion of contents.

% Usage: \Transpose { &-separated cells \\ &-separated cells \\ ...
%                     ...\\&-separated cells and *no* final \\ }

% Can be used inside amsmath matrices as well as inside LaTeX tabular's
% 
\def\Transpose #1{\romannumeral0\expandafter
                  \Mar@Transpose@a\romannumeral`^^@\Mar@DoOneRow #1\\!\\}

\def\Mar@DoOneRow #1\\{\Mar@DoOneRow@a {}#1&^^@&}%

\def\Mar@DoOneRow@a #1#2&{%
    \if^^@\detokenize{#2}\expandafter\@gobble\fi
    \Mar@DoOneRow@a {#1#2\\}%
}%

% unbraced #1 here will end in \\^^@\\
\def\Mar@Transpose@a #1#2\\{\ifx!#2\expandafter\Mar@FinishTranspose\fi
    \expandafter\Mar@Transpose@b\romannumeral`^^@\Mar@DoOneRow@a {}#2&^^@&#1}

% unbraced #1 here will end with \\^^@\\
% It represents a new transposed row with an extra ^^@ cell.
% The #2 here ends in \\. It holds the first already transposed rows.
\def\Mar@Transpose@b #1#2^^@\\{\Mar@Join {}#2^^@!#1}

% when #3=^^@ happens after #4\\ there is still ^^@\\
% update slightly simplifies \Mar@Join.
\def\Mar@Join #1#2\\#3!#4\\%
   {\if^^@\detokenize{#3}\expandafter\Mar@EndJoin\fi
    \Mar@Join {#1#2&#4\\}#3!}%

% unbraced #1 ends with \\
\def\Mar@EndJoin\Mar@Join #1^^@!^^@\\{\Mar@Transpose@a {#1^^@\\}}

\def\Mar@FinishTranspose
    #1&^^@&#2\\^^@\\{ #2}% the \\^^@\\ pattern helps removing a final \\

\catcode`! 12
\catcode0 15 % null character "invalid": this is default LaTeX setting.
\makeatother

\begin{document}

A bmatrix example:
\[
\begin{bmatrix}
    0 & -1 & -1 \\
   -1 & -1 & 0  \\
   -1 & 0  & 1  \\
    1 & 0  & 0  \\
    0 & 0  & -1 \\
   -1 & 2  & 1
\end{bmatrix}\longrightarrow
\begin{bmatrix}
  \Transpose {
    0 & -1 & -1 \\
   -1 & -1 & 0  \\
   -1 & 0  & 1  \\
    1 & 0  & 0  \\
    0 & 0  & -1 \\
   -1 & 2  & 1 }
\end{bmatrix}
\]

% works the same.
% \[
% \begin{pmatrix}
%   \Transpose {
%     0 & -1 & -1 \\
%    -1 & -1 & 0  \\
%    -1 & 0  & 1  \\
%     1 & 0  & 0  \\
%     0 & 0  & -1 \\
%    -1 & 2  & 1 }
% \end{pmatrix}
% \]

A tabular example:

\begin{center}
\begin{tabular}{c|>{\bfseries}c|>{\itshape}c}
  L1 & L2 & L3 \\
  M1 & M2 & M3 \\
  N1 & N2 & N3
\end{tabular}
${}\longrightarrow{}$
\begin{tabular}{c|>{\bfseries}c|>{\itshape}c}
  \Transpose{
  L1 & L2 & L3 \\
  M1 & M2 & M3 \\
  N1 & N2 & N3
}% <- this to avoid a spurious space which tabular does not remove
\end{tabular}
\end{center}

A macro-level example to illustrate that two expansion steps are enough and
that contents are not prematurely expanded:

% spaces from endlines and around tabs
% which disappear in tabular/matrix treatments
% are not trimmed by \Transpose itself 

\def\www {  
  \Arbitrary L1 & \Stuff L2 & \Here L3 \\
  M1 & M2 & M3 \\
  N1 & N2 & N3
}

\expandafter\expandafter\expandafter\def
\expandafter\expandafter\expandafter\zzz
\expandafter\expandafter\expandafter{\Transpose {
  \Arbitrary L1 & \Stuff L2 & \Here L3 \\
  M1 & M2 & M3 \\
  N1 & N2 & N3
}}

\ttfamily

\meaning\www

becomes

\meaning\zzz

% And finally a "write to file" example
\newwrite\testout

\immediate\openout\testout=\jobname-tranpose.out

\immediate\write\testout{\unexpanded\expandafter\expandafter\expandafter
   {\Transpose  {
  \Arbitrary L1 & \Stuff L2 & \Here L3 \\
  M1 & M2 & M3 \\
  N1 & N2 & N3
}}}

\end{document}

“filename-transpose.out”文件包含：

 \Arbitrary L1 & M1 & N1 \\ \Stuff L2 & M2 & N2 \\ \Here L3 & M3 & N3

Question 4

不是一个真正的 TeX 答案，但对某些人可能仍然有用。使用 Emacs，您可以从 LaTeX/AUCTeX 切换到 Calc 模式，执行矩阵运算并返回。

假设我们有一个要转置的 bmatrix

\[
\begin{bmatrix}
    0  & -1 & -1 \\
    -1 & -1 &  0 \\
    -1 &  0 &  1 \\
     1 &  0 &  0 \\
     0 &  0 & -1 \\
    -1 &  2 &  1 
\end{bmatrix}
\]

将光标放在环境内并按C-x * e进入 Calc 嵌入模式

\[
\begin{pmatrix} 0 & -1 & 
 -1 \\ -1 & -1 & 0 \\ -1 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \\ -1 & 2 & 1 \end{pmatrix}
\]

Calc 理解 LaTeX 语法，但搞乱了对齐，变成bmatrix了pmatrix。我们稍后再讨论这个问题。

按下d N可切换到正常的 Calc 显示模式，您可能会发现这种模式更适合工作

% [calc-mode: language: nil]
\[
[ [ 0,  -1, -1 ]
  [ -1, -1, 0  ]
  [ -1, 0,  1  ]
  [ 1,  0,  0  ]
  [ 0,  0,  -1 ]
  [ -1, 2,  1  ] ]
\]

v t转置

% [calc-mode: language: nil]
\[
[ [ 0,  -1, -1, 1, 0,  -1 ]
  [ -1, -1, 0,  0, 0,  2  ]
  [ -1, 0,  1,  0, -1, 1  ] ]
\]

C-2 d L让我们回到 LaTeX 语法，C-2是一个可选的前缀，将其格式化为具有正确对齐的二维矩阵

% [calc-mode: language: (latex 2)]
\[
\begin{pmatrix}
0 & -1 & -1 & 1 & 0 & -1 \\ 
-1 & -1 & 0 & 0 & 0 & 2 \\ 
-1 & 0 & 1 & 0 & -1 & 1
\end{pmatrix}
\]

C-x * e禁用 Calc 嵌入模式并让我们恢复正常编辑。现在唯一的问题是我们有一个pmatrix而不是原始的bmatrix。不幸的是，这是预料之中的，并记录在 Calc 文档中……但修复它很简单！如果您使用 AucTeX，只需按下C-u C-c C-e矩阵内部并输入bmatrix。

Answer