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在 LaTeX 中执行矩阵运算（加法、乘法、转置等）？

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在 LaTeX 中执行矩阵运算（加法、乘法、转置等）？

我知道该calc包可以在 LaTeX 中执行中缀表示法算术......但我想要更多！

我想在 LaTeX 中执行（不一定是用中缀表示法）线性代数运算，例如标量乘法、矩阵加法和乘积，然后在array或 的某个矩阵环境中打印结果amsmath。

为什么我要直接在 LaTeX 中执行此操作？为什么我不简单地使用一些线性代数软件，例如 Matlab、Mathematica 等？

好吧，假设我想通过许多数值示例向读者介绍一个详细的线性代数计算。当然，我可以先手动执行所有步骤，然后在输入文件中硬编码每个步骤的结果。但是，这种方法

  • 容易出错（LaTeX 的array排版不太方便用户使用），
  • 缺乏可维护性（我可能决定更改示例中的数据，这意味着我必须修改后续的所有内容）。

因此我的问题是：有没有办法在 LaTeX 中轻松地执行线性代数运算？

理想情况下，我希望

  1. 模仿 Matlab 定义矩阵的语法（使用逗号作为列分隔符，使用分号作为行尾字符），对其执行操作，提取子矩阵等。语法可能类似于以下内容：

    \let\A{1,2,3;4,5,6}
\let\b{1;0;0}
\let\c\matrixprod{\A,\b}
\let\d\submatrix{\c}{(2,1)}

  2. array有一个在或环境中排版“矩阵对象”的命令matrix，例如

    \typesetmatrix[bmatrix]{A}

  3. 能够对任意维度（尽管相对较小）的矩阵执行运算（编辑：不仅仅是包装中的 2x2 和 3x3 calculator）。

目前是否可以通过某些软件包实现这一点？如果不行，我正在考虑撸起袖子实施一些措施，但这可能相当困难，而且我想避免重新发明轮子 :)

编辑关于其他有用的操作：

  • diag（提取方阵的对角线）
  • 痕迹
  • 决定因素
  • 规范
  • 条件编号
  • 逆

甚至更高级的矩阵运算也很棒，但可能很难实现：

  • 特征值和特征向量，
  • QR、最小二乘法等。
  • 奇异值分解，
  • 其他常见的矩阵分解。

答案1

以下是一些用于操作任意大小矩阵的代码。目前，它可以执行加法、减法和乘法（以及获取单个条目和转置矩阵等）。条目是l3fp支持的浮点数（精度为 16 位）。

% Programming-level functions: \fpm_new:N, \fpm_set:Nn, \fpm_gset:Nn,
% \fpm_add:NNN, \fpm_sub:NNN, \fp_neg:NN, \fp_transpose:NN, \fp_mul:NNN.
%
% Expandable programming-level functions: \fpm_lines:N, \fpm_columns:N,
% \fpm_get:Nnn.
%
% Document-level functions: \matnew, \matset, \matgset, \matadd,
% \matsub, \matmul, \mattypeset.
%
\RequirePackage{expl3}
{
  \ExplSyntaxOn
  %
  % Programming-level code, for adding, multiplying, matrices.  A matrix
  % of size |MxN| is stored as a token list of the form
  %
  % \s__fpm { M } { N } { {a11} ... {a1N} } ... { {aM1} ... {aMN} } ;
  %
  % where |\s__fpm| is a marker used to recognize matrices, |M| and |N|
  % are non-negative integers, and |aij| are floating point numbers.
  %
  % (1) Variables.
  %
  \cs_new_eq:NN \s__fpm \scan_stop: % A marker.
  \tl_const:Nn \c_empty_fpm { \s__fpm { 0 } { 0 } ; }
  \cs_new_eq:NN \l__fpm_tmpa_fpm \c_empty_fpm
  \seq_new:N \l__fpm_lines_seq
  \int_new:N \l__fpm_lines_A_int
  \int_new:N \l__fpm_lines_B_int
  \int_new:N \l__fpm_columns_A_int
  \int_new:N \l__fpm_columns_B_int
  \tl_new:N \l__fpm_matrix_A_tl
  \tl_new:N \l__fpm_matrix_B_tl
  \tl_new:N \l__fpm_matrix_C_tl
  \seq_new:N \l__fpm_matrix_A_seq
  \seq_new:N \l__fpm_matrix_B_seq
  \seq_new:N \l__fpm_one_line_A_seq
  \seq_new:N \l__fpm_one_line_B_seq
  \tl_new:N \l__fpm_one_line_A_tl
  \int_new:N \l__fpm_tmpa_int
  %
  % (3) Storing matrices.
  %
  \cs_new_protected:Npn \fpm_new:N #1
    { \cs_new_eq:NN #1 \c_empty_fpm }
  \cs_new_protected_nopar:Npn \fpm_set:Nn
    { \__fpm_set:NNn \tl_set:Nx }
  \cs_new_protected_nopar:Npn \fpm_gset:Nn
    { \__fpm_set:NNn \tl_gset:Nx }
  \cs_new_protected:Npn \__fpm_set:NNn #1#2#3
    {
      \seq_set_split:Nnn \l__fpm_lines_seq { ; } {#3}
      \seq_set_filter:NNn \l__fpm_lines_seq \l__fpm_lines_seq
        { ! \tl_if_empty_p:n {##1} }
      %
      % Now all lines are non-empty.
      %
      \tl_clear:N \l__fpm_matrix_A_tl
      \int_zero:N \l__fpm_lines_A_int
      \int_zero:N \l__fpm_columns_A_int
      \seq_map_inline:Nn \l__fpm_lines_seq
        {
          \int_incr:N \l__fpm_lines_A_int
          \seq_set_from_clist:Nn \l__fpm_one_line_A_seq {##1}
          \int_set:Nn \l__fpm_tmpa_int { \seq_count:N \l__fpm_one_line_A_seq }
          \int_compare:nNnT \l__fpm_columns_A_int = \c_zero
            { \int_set_eq:NN \l__fpm_columns_A_int \l__fpm_tmpa_int }
          \int_compare:nNnF \l__fpm_tmpa_int = \l__fpm_columns_A_int
            { \seq_map_break:n { \msg_error:nn { fpm } { invalid-size } } }
          \tl_put_right:Nx \l__fpm_matrix_A_tl
            { { \seq_map_function:NN \l__fpm_one_line_A_seq \__fpm_set_aux:n } }
        }
      #1 #2
        {
          \s__fpm
          { \int_use:N \l__fpm_lines_A_int }
          { \int_use:N \l__fpm_columns_A_int }
          \l__fpm_matrix_A_tl
          ;
        }
    }
  \cs_new:Npn \__fpm_set_aux:n #1 { { \fp_to_tl:n {#1} } }
  %
  % (4) Extracting the size of a matrix, and its contents.
  % |#1| is the matrix, |#2|, |#3| integer variables receiving the
  % number of lines and of columns, and |#4| a token list receiving the
  % contents of the matrix.
  %
  \cs_new_protected:Npn \__fpm_get_parts:NNNN #1#2#3#4
    { \exp_after:wN \__fpm_get_parts:NnnwNNN #1 #2 #3 #4 }
  \cs_new_protected:Npn \__fpm_get_parts:NnnwNNN \s__fpm #1#2#3 ; #4#5#6
    {
      \int_set:Nn #4 {#1}
      \int_set:Nn #5 {#2}
      \tl_set:Nn #6 {#3}
    }
  %
  % (5) Some expandable functions: getting one entry, getting the size.
  %
  \cs_new:Npn \fpm_lines:N #1
    { \exp_after:wN \__fpm_lines:NnnwN #1 \use_i:nn }
  \cs_new:Npn \fpm_columns:N #1
    { \exp_after:wN \__fpm_lines:NnnwN #1 \use_ii:nn }
  \cs_new:Npn \__fpm_lines:NnnwN \s__fpm #1#2#3 ; #4 { #4 {#1} {#2} }
  \cs_new:Npn \fpm_get:Nnn #1#2#3
    { \exp_after:wN \__fpm_get:Nnnwnn #1 #2 #3 }
  \cs_new:Npn \__fpm_get:Nnnwnn \s__fpm #1#2#3 ; #4#5
    { \exp_args:Nf \tl_item:nn { \tl_item:nn {#3} {#4} } {#5} }
  %
  % (6) Summing matrices
  %
  \cs_new_protected_nopar:Npn \fpm_add:NNN { \__fpm_add:NNNN + }
  \cs_new_protected_nopar:Npn \fpm_sub:NNN { \__fpm_add:NNNN - }
  \cs_new_protected:Npn \__fpm_add:NNNN #1#2#3#4
    {
      \tl_set:Nn \l__fpm_sign_tl {#1}
      \__fpm_get_parts:NNNN #3
        \l__fpm_lines_A_int \l__fpm_columns_A_int \l__fpm_matrix_A_tl
      \__fpm_get_parts:NNNN #4
        \l__fpm_lines_B_int \l__fpm_columns_B_int \l__fpm_matrix_B_tl
      \int_compare:nNnTF \l__fpm_lines_A_int = \l__fpm_lines_B_int
        {
          \int_compare:nNnTF \l__fpm_columns_A_int = \l__fpm_columns_B_int
            { \__fpm_add:N #2 }
            { \msg_error:nn { fpm } { invalid-size } }
        }
        { \msg_error:nn { fpm } { invalid-size } }
    }
  \cs_new_protected:Npn \__fpm_add:N #1
    {
      \seq_set_split:NnV \l__fpm_matrix_A_seq { } \l__fpm_matrix_A_tl
      \seq_set_split:NnV \l__fpm_matrix_B_seq { } \l__fpm_matrix_B_tl
      \tl_clear:N \l__fpm_matrix_C_tl
      \seq_mapthread_function:NNN
        \l__fpm_matrix_A_seq
        \l__fpm_matrix_B_seq
        \__fpm_add_lines:nn
      \tl_set:Nx #1
        {
          \s__fpm
          { \int_use:N \l__fpm_lines_A_int }
          { \int_use:N \l__fpm_columns_A_int }
          \l__fpm_matrix_C_tl
          ;
        }
    }
  \cs_new_protected:Npn \__fpm_add_lines:nn #1#2
    {
      \seq_set_split:Nnn \l__fpm_one_line_A_seq { } {#1}
      \seq_set_split:Nnn \l__fpm_one_line_B_seq { } {#2}
      \tl_put_right:Nx \l__fpm_matrix_C_tl
        {
          {
            \seq_mapthread_function:NNN
              \l__fpm_one_line_A_seq
              \l__fpm_one_line_B_seq
              \__fpm_add_entries:nn
          }
        }
    }
  \cs_new:Npn \__fpm_add_entries:nn #1#2
    { { \fp_to_tl:n { #1 \l__fpm_sign_tl #2 } } }
  %
  % (7) Negating all entries.
  %
  \cs_new_protected:Npn \fpm_neg:NN #1#2
    { \tl_set:Nx #1 { \exp_after:wN \__fpm_neg:Nnnw #2 } }
  \cs_new:Npn \__fpm_neg:Nnnw \s__fpm #1#2#3 ;
    { \s__fpm {#1} {#2} \tl_map_function:nN {#3} \__fpm_neg_aux:n ; }
  \cs_new:Npn \__fpm_neg_aux:n #1
    { { \tl_map_function:nN {#1} \__fpm_neg_auxii:n } }
  \cs_new:Npn \__fpm_neg_auxii:n #1
    { { \fp_to_tl:n { - #1 } } }
  %
  % (8) Transposing a matrix.
  %
  \cs_new_protected:Npn \fpm_transpose:NN #1#2
    {
      \__fpm_get_parts:NNNN #2
        \l__fpm_lines_A_int \l__fpm_columns_A_int \l__fpm_matrix_A_tl
      \seq_set_split:NnV \l__fpm_matrix_A_seq { } \l__fpm_matrix_A_tl
      \tl_clear:N \l__fpm_matrix_B_tl
      \prg_replicate:nn { \l__fpm_columns_A_int }
        {
          \tl_put_right:Nx \l__fpm_matrix_B_tl
            { { \seq_map_function:NN \l__fpm_matrix_A_seq \__fpm_wrap_head:n } }
          \seq_set_map:NNn \l__fpm_matrix_A_seq \l__fpm_matrix_A_seq
            { \tl_tail:n {##1} }
        }
      \tl_set:Nx #1
        {
          \s__fpm
          { \int_use:N \l__fpm_columns_A_int }
          { \int_use:N \l__fpm_lines_A_int }
          \l__fpm_matrix_B_tl
          ;
        }
    }
  \cs_new:Npn \__fpm_wrap_head:n #1 { { \tl_head:n {#1} } }
  %
  % (9) Multiplying matrices.
  %
  \cs_new_protected:Npn \fpm_mul:NNN #1#2#3
    {
      \int_compare:nNnTF { \fpm_columns:N #2 } = { \fpm_lines:N #3 }
        {
          \fpm_transpose:NN \l__fpm_tmpa_fpm #3
          \__fpm_get_parts:NNNN #2
            \l__fpm_lines_A_int \l__fpm_columns_A_int \l__fpm_matrix_A_tl
          \__fpm_get_parts:NNNN #3
            \l__fpm_lines_B_int \l__fpm_columns_B_int \l__fpm_matrix_B_tl
          \tl_set:Nx #1
            {
              \s__fpm
              { \int_use:N \l__fpm_lines_A_int }
              { \int_use:N \l__fpm_columns_B_int }
              \tl_map_function:NN \l__fpm_matrix_A_tl \__fpm_mul_line:n
              ;
            }
        }
        { \msg_error:nn { fpm } { invalid-size } }
    }
  \cs_new:Npn \__fpm_mul_line:n #1
    { { \exp_after:wN \__fpm_mul_line:Nnnwn \l__fpm_tmpa_fpm {#1} } }
  \cs_new:Npn \__fpm_mul_line:Nnnwn \s__fpm #1#2#3 ; #4
    { \__fpm_mul_line:nn {#4} #3 \q_recursion_tail \q_recursion_stop }
  \cs_new:Npn \__fpm_mul_line:nn #1#2
    {
      \quark_if_recursion_tail_stop:n {#2}
      {
        \fp_to_tl:n
          {
            \__fpm_mul_one:nwn #1 \use_none_delimit_by_q_stop:w
              \q_mark #2 \q_nil \q_stop
            0
          }
      }
      \__fpm_mul_line:nn {#1}
    }
  \cs_new:Npn \__fpm_mul_one:nwn #1#2 \q_mark #3
    { #1 * #3 + \__fpm_mul_one:nwn #2 \q_mark }
  %
  %
  % Messages.
  %
  \msg_new:nnn { fpm } { invalid-size }
    { Sizes~of~matrices~or~lines~don't~match. }
}
\RequirePackage{amsmath, siunitx}
{
  \ExplSyntaxOn
  %
  % Turning matrices into arrays for display.
  %
  \cs_new_protected:Npn \fpm_to_array:N #1
    {
      \begin{pmatrix}
        \exp_after:wN \__fpm_to_array:Nnnw #1
      \end{pmatrix}
    }
  \cs_new_eq:NN \__fpm_newline: ? % Dummy def.
  \cs_new_protected:Npn \__fpm_to_array:Nnnw \s__fpm #1#2#3 ;
    {
      \cs_gset_nopar:Npn \__fpm_newline:
        { \cs_gset_nopar:Npn \__fpm_newline: { \\ } }
      \tl_map_inline:nn {#3}
        {
          \__fpm_newline:
          \seq_set_split:Nnn \l__fpm_one_line_A_seq { } {##1}
          \seq_set_map:NNn \l__fpm_one_line_A_seq \l__fpm_one_line_A_seq
            { \__fpm_to_array_entry:n {####1} }
          \seq_use:Nnnn \l__fpm_one_line_A_seq { & } { & } { & }
        }
    }
  \cs_new_protected:Npn \__fpm_to_array_entry:n #1
    {
      \str_case:nnn {#1}
        {
          { nan } { \text{nan} }
          { inf } { \infty }
          { -inf } { -\infty }
        }
        { \num{#1} }
    }
}

\RequirePackage{xparse}
\ExplSyntaxOn
%
% Document-level functions.
%
\NewDocumentCommand { \matnew } { m } { \fpm_new:N #1 }
\NewDocumentCommand { \matset } { mm } { \fpm_set:Nn #1 {#2} }
\NewDocumentCommand { \matgset } { mm } { \fpm_gset:Nn #1 {#2} }
\NewDocumentCommand { \matadd } { mmm } { \fpm_add:NNN #1 #2 #3 }
\NewDocumentCommand { \matsub } { mmm } { \fpm_sub:NNN #1 #2 #3 }
\NewDocumentCommand { \matneg } { mm } { \fpm_neg:NN #1 #2 }
\NewDocumentCommand { \mattranspose } { mm } { \fpm_transpose:NN #1 #2 }
\NewDocumentCommand { \matmul } { mmm } { \fpm_mul:NNN #1 #2 #3 }
\NewDocumentCommand { \mattypeset } { m }
  { \fpm_to_array:N #1 }
\DeclareExpandableDocumentCommand { \matget } { mmm }
  { \fp_to_tl:n { \fpm_get:Nnn #1 {#2} {#3} } }
\ExplSyntaxOff

\documentclass{article}
\begin{document}
  \matnew \X
  \matnew \Y
  \matnew \Z
  \matset \X { 1 , 2 + 3 ; 4 , 3.4e22 }
  \matset \Y { 3 , 4 ; -5 , 6 }
  \begin{align}
    \matadd \Z \X \Y
    \mattypeset \Z & = \mattypeset \X + \mattypeset \Y \\
    \matsub \Z \X \Y
    \mattypeset \Z & = \mattypeset \X - \mattypeset \Y \\
    \matmul \Z \X \Y
    \mattypeset \Z & = \mattypeset \X \times \mattypeset \Y \\
    \matmul \Z \Y \X
    \mattypeset \Z & = \mattypeset \Y \times \mattypeset \X
  \end{align}
  \(X[1,2] = \matget\X{1}{2}\).
\end{document}

编辑：添加\matget，提取矩阵中的单个条目。

在此处输入图片描述

答案2

calculator包可能会有帮助。

在此处输入图片描述

在此处输入图片描述

答案3

矩阵运算（乘法、逆运算、行列式）可以精确计算或使用浮点数（任意精度）

这个 2013 年 11 月的答案于 2017 年 3 月进行了编辑，因为早在 2014 年底，从 xint 中删除宏就导致代码无法使用，并且自 1.1（2014/10/28）以来 xintfrac 不会自动加载 xinttools。

\documentclass{article}
\usepackage[paperheight=100cm,vscale=0.9]{geometry}
\usepackage{xintfrac}
\RequirePackage{array} 
% november 8-11, 2013
\catcode`_ 11

% update 2017/03/23, because some macros stopped being defined by later
% versions of xint... 

% A. xintfrac stopped loading xinttools at 1.1 (2014/10/28)
\usepackage{xinttools}

% B. \XINTinFloatSum got removed at 1.1a (2014/11/07)

% \lverb|1.09a: quick write-up, for use by \xintfloatexpr, will need to be
% thought through  again. Renamed (and slightly modified) in 1.09h. Should be
% extended for optional precision. Should be rewritten for optimization. |
\def\XINTinFloatSum   {\romannumeral0\XINTinfloatsum }%
\def\XINTinfloatsum #1{\expandafter\XINT_floatsum_a\romannumeral-`0#1\relax }%
\def\XINT_floatsum_a #1{\expandafter\XINT_floatsum_b
                        \romannumeral0\XINTinfloat[\XINTdigits]{#1}\Z }%
\def\XINT_floatsum_b #1\Z #2%
           {\expandafter\XINT_floatsum_c\romannumeral-`0#2\Z {#1}\Z}%
\def\XINT_floatsum_c #1%
           {\xint_gob_til_relax #1\XINT_floatsum_e\relax\XINT_floatsum_d #1}%
\def\XINT_floatsum_d #1\Z
           {\expandafter\XINT_floatsum_b\romannumeral0\XINTinfloatadd {#1}}%
\def\XINT_floatsum_e #1\Z #2\Z { #2}%

% C. \XINT_Abs got removed at 1.2i (2016/12/13)
\def\XINT_Abs #1{\romannumeral0\XINT_abs #1}%
% end of update

\makeatletter

\let\MAT_xintfloatsum\XINTinFloatSum

\newcount\MAT_cnta
\newcount\MAT_cntb
\newcount\MAT_cntc
\newcount\MAT_cntd
\newcount\MAT_cnte

\def\MATset      {\def\MAT_xintin {\xintRaw}\MATset_ }%
\def\MATfloatset {\def\MAT_xintin {\XINTinFloat [\XINTdigits]}\MATset_ }%

\def\MATset_ #1#2{%
    \def\MATset_name{#1}%
    \edef\MAT_tmpa {#2}%
    \MAT_cnta \xint_c_ % sets \MAT_cnta to zero
    \expandafter\MATset_a 
    \romannumeral0\expandafter\xintzapspaces\expandafter{\MAT_tmpa};!;%
}%
\def\MATset_a {\futurelet\XINT_token\MATset_b }%
\def\MATset_b #1;{\def\MAT_tmpa{#1}%
                  \ifx\XINT_token;\expandafter\MATset_w
                  \else
                  \ifx\XINT_token!%
                          \expandafter\expandafter\expandafter\MATset_x
                     \else
                          \expandafter\expandafter\expandafter\MATset_c
                  \fi\fi }%
\def\MATset_w !;{\MATset_x }%
\def\MATset_x {\expandafter\def
  \csname MAT@\expandafter\string\MATset_name {I}\expandafter\endcsname
  \expandafter {\the\MAT_cnta }%
               \expandafter\def
  \csname MAT@\expandafter\string\MATset_name {J}\expandafter\endcsname 
  \expandafter {\the\MAT_cntb }%
  \expandafter\edef \MATset_name [##1]%
     {\noexpand\csname MAT@\expandafter\string\MATset_name 
               \noexpand\MAT_in ##1,\noexpand\xint_bye,\endcsname }%
}%
%
\def\MAT_in #1,#2,{\xint_bye #2\xint_gobble_iv\xint_bye
                   {\the\numexpr #1}{\the\numexpr #2}\xint_gobble_iii 
                   {\xintZapSpaces{#1}}}%
%
\def\MATset_c {\advance\MAT_cnta \xint_c_i % row count ++
               \MAT_cntb \xint_c_ % column count intially zero
               \expandafter\MATset_d\romannumeral0\expandafter
               \xintzapspaces\expandafter {\MAT_tmpa},!,}%
\def\MATset_d {\futurelet\XINT_token\MATset_e }%
\def\MATset_e #1,{\ifx\XINT_token!\expandafter\MATset_a
  \else
      \advance\MAT_cntb \xint_c_i
      \expandafter\def
  \csname MAT@\expandafter\string\MATset_name 
            {\the\MAT_cnta}{\the\MAT_cntb}\expandafter\endcsname
  \expandafter{\romannumeral-`0\MAT_xintin{\xintZapSpacesB{#1}}}%
  \expandafter\MATset_d\fi
}%

% \MATdef

\def\MATdef      {\def\MAT_xintin {\xintRaw}%
                  \MATdef_ }%
\def\MATfloatdef {\def\MAT_xintin {\XINTinFloat [\XINTdigits]}%
                  \MATdef_ }%

% #3 should be a replacement text with #1 and #2 for horizontal and vertical
% indices, which can be expanded to its final result inside an \edef, and this
% result must be parsable by the xint macros. 

% WARNING! version of NOV 10 defined only square matrices, this one of NOV 11
% defines *rectangular matrices and has one more argument*

\def\MATdef_ #1#2#3#4{%
    \MAT_cnta #2\relax
    \MAT_cntb #3\relax
    \def\MAT_tmpa ##1##2{#4}%
    \MAT_cntc \xint_c_i % =1
    \xintloop
      {\expandafter\def\expandafter\MAT_tmpc\expandafter 
                           {\expandafter{\the\MAT_cntc}}%
       \MAT_cntd \xint_c_i %=1
       \xintloop
         \expandafter\def\expandafter\MAT_tmpd\expandafter 
                            {\expandafter{\the\MAT_cntd}}%
         \edef\MAT_tmpb {\expandafter\expandafter\expandafter\MAT_tmpa 
                         \expandafter\MAT_tmpc\MAT_tmpd}%
         \expandafter\def
         \csname MAT@\string#1\MAT_tmpc\MAT_tmpd\expandafter\endcsname 
         \expandafter {\romannumeral-`0\MAT_xintin 
                 {\expandafter\xintZapSpacesB\expandafter{\MAT_tmpb}}}%
       \ifnum\MAT_cntd<\MAT_cntb
         \advance\MAT_cntd \xint_c_i
       \repeat
     \ifnum\MAT_cntc<\MAT_cnta
        \advance\MAT_cntc \xint_c_i
    }\repeat
    \expandafter\def
    \csname MAT@\string#1{I}\expandafter\endcsname\expandafter {\the\MAT_cnta}%
    \expandafter\def
    \csname MAT@\string#1{J}\expandafter\endcsname\expandafter {\the\MAT_cntb}%
    \edef #1[##1]%
       {\noexpand\csname 
         MAT@\string#1\noexpand\MAT_in ##1,\noexpand\xint_bye,\endcsname }%
}%

% \MATsetentry
\def\MATsetentry      {\def\MAT_xintin {\xintRaw}%
                       \MATsetentry_ }%
\def\MATfloatsetentry {\def\MAT_xintin 
                             {\XINTinFloat [\XINTdigits]}%
                       \MATsetentry_ }%

\def\MATsetentry_ #1#2#3{%
    \edef\MAT_tmpa {#3}%
    \expandafter\def
    \csname MAT@\string#1\MAT_in #2,\xint_bye,\expandafter\endcsname\expandafter
    {\romannumeral-`0\MAT_xintin 
     {\expandafter\xintZapSpaces\expandafter{\MAT_tmpa}}}%
}%


% NOTA BENE
% use of \xintFor is for ease of coding. In an official package, I would use
% special loops for optimal efficiency (the \xintFor is a general tool which has
% safeguards against situations which do not arise here, like groups suddenly
% closing)

% 10 november:
% Current version has already replaced use of \xintFor by \xintloop in a number
% of places, notably for the computation of inverses and determinants.

% but I leave \xintFor in a number of macros.
% Improvements from using less \edef's in various places

\def\MATrelax #1{%
    \toks2 \expandafter {\romannumeral-`0\xintSeq {1}{#1[I]}}%
    \toks4 \expandafter {\romannumeral-`0\xintSeq {1}{#1[J]}}%
    \xintFor* ##1 in {\the\toks2 }
    \do{\xintFor* ##2 in {\the\toks4 }
        \do{\expandafter\let\csname MAT@\string#1{##1}{##2}\endcsname\relax }}%
  \expandafter\let\csname MAT@\string#1{I}\endcsname \relax
  \expandafter\let\csname MAT@\string#1{J}\endcsname \relax
  \let #1\relax
}%

\def\MATlet #1#2{%
    \toks2 \expandafter {\romannumeral-`0\xintSeq {1}{#2[I]}}%
    \toks4 \expandafter {\romannumeral-`0\xintSeq {1}{#2[J]}}%
    \xintFor* ##1 in {\the\toks2 }
    \do{\xintFor* ##2 in {\the\toks4 }
        \do{\expandafter\let
     \csname MAT@\string#1{##1}{##2}\expandafter\endcsname
     \csname MAT@\string#2{##1}{##2}\endcsname
     }}%
  \expandafter\edef\csname MAT@\string#1{I}\endcsname {#2[I]}%
  \expandafter\edef\csname MAT@\string#1{J}\endcsname {#2[J]}%
  \edef #1[##1]%
     {\noexpand\csname 
      MAT@\string#1\noexpand\MAT_in ##1,\noexpand\xint_bye,\endcsname }%
}%

% \MATapply
% argument #1 is \macro or \macro {arg1}..{argn} where \macro is a macro with
% n+1 arguments.

\def\MATapply #1#2{%
    \toks2 \expandafter {\romannumeral-`0\xintSeq {1}{#2[I]}}%
    \toks4 \expandafter {\romannumeral-`0\xintSeq {1}{#2[J]}}%
    \xintFor* ##1 in {\the\toks2 }
    \do{\xintFor* ##2 in {\the\toks4 }
        \do
        {\toks@ {#1}%
         \expandafter\edef
        \csname MAT@\string#2{##1}{##2}\expandafter\expandafter\expandafter
        \endcsname\expandafter\expandafter\expandafter
        {\expandafter\the\expandafter\toks@\expandafter
             {\romannumeral-`0\csname MAT@\string#2{##1}{##2}\endcsname }}%
        }%
    }%
}%

% TRANSPOSE
% Code rewritten to illustrate how one can proceed with \xintloop and counts
% rather than \xintFor. 
\def\MATtranspose #1#2{%
    \MAT_cnta #2[I]\relax
    \MAT_cntb #2[J]\relax
    \MAT_cntd \xint_c_i
    \xintloop {%
      \toks0 \expandafter{\the\MAT_cntd}%
      \MAT_cnte \xint_c_i
      \xintloop
         \toks2 \expandafter{\the\MAT_cnte}%
         \expandafter\let
         \csname MAT@_tmp{\the\toks2}{\the\toks0}\expandafter\endcsname
         \csname MAT@\string#2{\the\toks0}{\the\toks2}\endcsname 
         \ifnum \MAT_cnte < \MAT_cntb \advance\MAT_cnte \xint_c_i
      \repeat
      \ifnum \MAT_cntd < \MAT_cnta \advance\MAT_cntd \xint_c_i
    }\repeat
    \MAT_cntd \xint_c_i
    \xintloop {%
      \toks0 \expandafter{\the\MAT_cntd}%
      \MAT_cnte \xint_c_i
      \xintloop
         \toks2 \expandafter{\the\MAT_cnte}%
         \expandafter\let
         \csname MAT@\string#1{\the\toks0}{\the\toks2}\expandafter\endcsname
         \csname MAT@_tmp{\the\toks0}{\the\toks2}\endcsname 
         \ifnum \MAT_cnte < \MAT_cnta \advance\MAT_cnte \xint_c_i
      \repeat
      \ifnum \MAT_cntd < \MAT_cntb \advance\MAT_cntd \xint_c_i
    }\repeat
  \expandafter\def\csname MAT@\string#1{I}\expandafter\endcsname
         \expandafter {\the\MAT_cntb }%
  \expandafter\def\csname MAT@\string#1{J}\expandafter\endcsname
         \expandafter {\the\MAT_cnta }%
  \edef #1[##1]%
     {\noexpand\csname 
      MAT@\string#1\noexpand\MAT_in ##1,\noexpand\xint_bye,\endcsname }%
}%

% SCALAR MULTIPLICATION

\def\MATsmul {\def\MAT_xintin {\xintRaw}%
              \def\MAT_MUL {\xintMul}%
              \MATsmul_ }%
\def\MATfloatsmul {\def\MAT_xintin {\XINTinFloat [\XINTdigits]}%
                   \def\MAT_MUL {\XINTinFloatMul}%
                   \MATsmul_ }%
\def\MATsmul_ #1#2#3{%
    \edef\MAT_tmpa {#2}%
    \expandafter\def\expandafter\MAT_tmpa\expandafter
    {\romannumeral-`0\MAT_xintin 
      {\expandafter\xintZapSpaces\expandafter{\MAT_tmpa}}}%
    \toks0 \expandafter {\romannumeral-`0\xintSeq {1}{#3[I]}}%
    \toks2 \expandafter {\romannumeral-`0\xintSeq {1}{#3[J]}}%
    \xintFor* ##1 in {\the\toks0 }
    \do{\xintFor* ##2 in {\the\toks2 }
         \do{\expandafter
             \def\csname MAT@\string#1{##1}{##2}\expandafter\endcsname 
             \expandafter{\romannumeral-`0\MAT_MUL\MAT_tmpa {#3[##1,##2]}}%
        }%
      }%
  \expandafter\edef\csname MAT@\string#1{I}\endcsname {#3[I]}%
  \expandafter\edef\csname MAT@\string#1{J}\endcsname {#3[J]}%
  \edef #1[##1]%
     {\noexpand\csname 
      MAT@\string#1\noexpand\MAT_in ##1,\noexpand\xint_bye,\endcsname }%
}%

% ADDITION

\def\MATadd {\def\MAT_ADD ##1##2{\xintIrr {\xintAdd {##1}{##2}}[0]}%
             \MATadd_ }%

\def\MATfloatadd {\def\MAT_ADD {\XINTinFloatAdd }\MATadd_ }%

\def\MATadd_ #1#2#3{%
    \edef\MAT_tmpa {\xintSeq {1}{#2[I]}}%
    \edef\MAT_tmpb {\xintSeq {1}{#2[J]}}%
    \xintFor* ##1 in \MAT_tmpa
    \do{\xintFor* ##2 in \MAT_tmpb
         \do{\expandafter\def\csname MAT@_tmp{##1}{##2}\expandafter\endcsname 
             \expandafter{\romannumeral-`0\MAT_ADD {#2[##1,##2]}{#3[##1,##2]}}%
         }%
      }%
   \xintFor* ##1 in \MAT_tmpa
   \do{\xintFor* ##2 in \MAT_tmpb
        \do{\expandafter\let
            \csname MAT@\string#1{##1}{##2}\expandafter\endcsname 
            \csname MAT@_tmp{##1}{##2}\endcsname 
           }%
       }%   
  \expandafter\edef\csname MAT@\string#1{I}\endcsname {#2[I]}%
  \expandafter\edef\csname MAT@\string#1{J}\endcsname {#2[J]}%
  \edef #1[##1]%
     {\noexpand\csname 
       MAT@\string#1\noexpand\MAT_in ##1,\noexpand\xint_bye,\endcsname }%
}%

% MULTIPLICATION

\def\MATmul {\def\MAT_MUL    {\xintMul }%
             \def\MAT_SUM ##1{\xintIrr {\xintSum {##1}}[0]}%
             \MATmul_ }%

\def\MATfloatmul  {\def\MAT_MUL {\XINTinFloatMul}%
                   \def\MAT_SUM {\MAT_xintfloatsum}%
                   \MATmul_ }%

\def\MATmul_ #1#2#3{%
    \edef\MAT_tmpa {\xintSeq {1}{#2[I]}}%
    \edef\MAT_tmpb {\xintSeq {1}{#3[J]}}%
    \edef\MAT_tmpc {\xintSeq {1}{#2[J]}}%
    \xintFor* ##1 in \MAT_tmpa
    \do{\xintFor* ##2 in \MAT_tmpb
         \do{%
            \def\MAT_tmpd ####1{\MAT_MUL {#2[##1,####1]}{#3[####1,##2]}}%
            \expandafter
                \def\csname MAT@_tmp{##1}{##2}\expandafter\endcsname 
            \expandafter
            {\romannumeral-`0\MAT_SUM{\xintApply\MAT_tmpd\MAT_tmpc}}%
         }%
      }%
   \xintFor* ##1 in \MAT_tmpa
   \do{\xintFor* ##2 in \MAT_tmpb
        \do{\expandafter\let
            \csname MAT@\string#1{##1}{##2}\expandafter\endcsname 
            \csname MAT@_tmp{##1}{##2}\endcsname 
           }%
       }%   
  \expandafter\edef\csname MAT@\string#1{I}\endcsname {#2[I]}%
  \expandafter\edef\csname MAT@\string#1{J}\endcsname {#3[J]}%
  \edef #1[##1]%
     {\noexpand\csname 
      MAT@\string#1\noexpand\MAT_in ##1,\noexpand\xint_bye,\endcsname }%
}%

% IDENTITY MATRIX
\def\MATid      {\def\MAT_tmpf{/1}\MAT_id }%
\def\MATfloatid {\def\MAT_tmpf{}\MAT_id }%
\def\MAT_id #1#2{%
    \MAT_cntc #2\relax
    \MAT_cnta \xint_c_i % 1
    \xintloop
      {\expandafter\def\expandafter\MAT_tmpa \expandafter{\the\MAT_cnta}%
       \MAT_cntb \xint_c_i % 1
       \xintloop
         \expandafter\edef
         \csname MAT@\string#1{\MAT_tmpa}{\the\MAT_cntb}\endcsname 
           {\ifnum\MAT_cntb=\MAT_cnta 1\else 0\fi \MAT_tmpf[0]}%
       \ifnum\MAT_cntb<\MAT_cntc
         \advance\MAT_cntb \xint_c_i
       \repeat
     \ifnum\MAT_cnta<\MAT_cntc
        \advance\MAT_cnta \xint_c_i
    }\repeat
    \expandafter\def\csname MAT@\string#1{I}\expandafter\endcsname
            \expandafter {\the\MAT_cntc}%
    \expandafter\def\csname MAT@\string#1{J}\expandafter\endcsname 
            \expandafter {\the\MAT_cntc}%
    \edef #1[##1]%
       {\noexpand\csname 
         MAT@\string#1\noexpand\MAT_in ##1,\noexpand\xint_bye,\endcsname }%
}%

% INVERSES AND DETERMINANTS

\def\MATinv {\def\MATinvordet_Ia{\MATinv_Ia}%
             \def\MATinvordet_II{\MATinv_II}%
             \MATinvordet }

\def\MATdet {\def\MAT_det {1/1[0]}% initial value
             \def\MATinvordet_Ia{\MATdet_Ia}%
             \def\MATinvordet_II{\edef\MAT_det{\xintIrr{\MAT_det}[0]}%
                                 \MATdet_end}%
             \MATinvordet }

\def\MATfloatinv {\def\MATinvordet_Ia{\MATinv_Ia}%
             \def\MATinvordet_II{\MATinv_II}%
             \MATfloatinvordet }

\def\MATfloatdet {\def\MAT_det {1[0]}% initial value
            \def\MATinvordet_Ia{\MATdet_Ia}%
            \def\MATinvordet_II{\edef\MAT_det{\xintFloat{\MAT_det}}\MATdet_end}%
            \MATfloatinvordet }

\def\MATandinverse #1{\def\MAT_name {#1}\MATinv_II }%

\def\MATinvordet #1#2{%
    \def\MAT_ZERO {0/1[0]}%
    \def\MAT_DIV ##1##2{\xintIrr{\xintDiv {##1}{##2}}}%
    \def\MAT_SUB ##1##2{\xintIrr{\xintSub {##1}{##2}}}%
    \def\MAT_MUL {\xintMul }%
    \MATid \MAT_invN {#2[I]}%
    \MATlet\MAT_invM  #2%
    \def\MAT_name {#1}%
    % \MAT_cntc is the size of the matrix. Will NOT be changed in subroutines.
    \MAT_cntc  #2[I]\relax
    \MAT_cnta \xint_c_i
    \MATinvordet_I 
}%
\def\MATfloatinvordet #1#2{%
    \def\MAT_ZERO {0.e0}%
    \def\MAT_DIV {\XINTinFloatDiv }%
    \def\MAT_SUB {\XINTinFloatSub }%
    \def\MAT_MUL {\XINTinFloatMul }%
    \MATfloatid \MAT_invN {#2[I]}%
    \MATlet\MAT_invM  #2%
    \def\MAT_name {#1}%
    % \MAT_cntc is the size of the matrix. Will NOT be changed in subroutines.
    \MAT_cntc  #2[I]\relax
    \MAT_cnta \xint_c_i
    \MATinvordet_I 
}%
\def\MATinvordet_I {\ifnum\MAT_cnta>\MAT_cntc 
                    \expandafter\MATinvordet_II
               \else\expandafter\MATinvordet_Ia
               \fi }%

\def\MATinv_II {\ifnum\MAT_cnta=\xint_c_i 
                     \expandafter\MATinv_end
                \else\expandafter\MATinv_IIa
                \fi }%
\def\MATinv_end {\expandafter\MATlet\MAT_name\MAT_invN }%
\def\MATdet_end {\expandafter\let\MAT_name\MAT_det }

\catcode`! 11
\def\MATinv_Ia {%    
    \MAT_cntb \MAT_cnta\relax
    \xintloop
       \xintifZero {\MAT_invM [\MAT_cntb,\MAT_cnta]}
       {\advance\MAT_cntb \xint_c_i 
        \ifnum\MAT_cntb>\MAT_cntc \MATinv_!\MATinvordet_I\fi
        \iftrue}
       {\iffalse}%
    \repeat
    \MATinv_Ipivot
    \ifnum\MAT_cntb>\MAT_cnta \MATinv_exc\fi
    \advance\MAT_cnta \xint_c_i
    \MATinvordet_I
}%
\def\MATdet_Ia {%    
    \MAT_cntb \MAT_cnta\relax
    \xintloop
       \xintifZero {\MAT_invM [\MAT_cntb,\MAT_cnta]}
       {\advance\MAT_cntb \xint_c_i 
        \ifnum\MAT_cntb>\MAT_cntc \MATdet_!\MATinvordet_I\fi
        \iftrue}
       {\iffalse}%
    \repeat
    \MATinv_Ipivot
    \ifodd\numexpr\MAT_cntb-\MAT_cnta\relax
          \edef\MAT_det{\xintOpp {\MAT_det}}%
    \fi
    \edef\MAT_det {\MAT_MUL {\MAT_pivot}{\MAT_det}}% 
    \ifnum\MAT_cntb>\MAT_cnta \MATinv_exc\fi
    \advance\MAT_cnta \xint_c_i
    \MATinvordet_I
}%
\def\MATinv_! #1\fi{\fi 
                   \xintbreakloopanddo
                   {NOT INVERTIBLE \on@line\typeout{NOT INVERTIBLE \on@line}%
                   \MATinv_end \def\MAT_tmpa ##1#1{}\MAT_tmpa }%
                   }%
\def\MATdet_! #1\fi{\fi 
                   \xintbreakloopanddo
                   {\edef\MAT_det{\MAT_ZERO}%
                    \MATdet_end \def\MAT_tmpa ##1#1{}\MAT_tmpa }%
                   }%
\catcode`! 12

\def\MATinv_IIa {%
    \advance\MAT_cnta -\xint_c_i 
    \MATinv_IIpivot
    \MATinv_II
}%

\def\MATinv_exc {%
% we optimize as we only need to do in M the indices > \MAT_cnta
% and in N the indices at most \MAT_cntb
    \toks0 \expandafter{\the\MAT_cnta}%
    \toks2 \expandafter{\the\MAT_cntb}%
% first we do in matrix M, column indices > "a"
    \MAT_cntd \MAT_cnta
    \xintloop
    \ifnum \MAT_cntd<\MAT_cntc
      \advance \MAT_cntd \xint_c_i
      \toks4 \expandafter{\the\MAT_cntd}%
      \expandafter\def\expandafter\MAT_tmpd\expandafter
        {\csname MAT@\string\MAT_invM{\the\toks0}{\the\toks4}\endcsname }%
      \expandafter\def\expandafter\MAT_tmpe\expandafter
        {\csname MAT@\string\MAT_invM{\the\toks2}{\the\toks4}\endcsname }%
      \expandafter\let\expandafter\MAT_tmpc\MAT_tmpd
      \expandafter\expandafter\expandafter\let\expandafter\MAT_tmpd\MAT_tmpe
      \expandafter\let\MAT_tmpe\MAT_tmpc
    \repeat
% Then we do in matrix N, column indices <= "b"
    \MAT_cntd \xint_c_i % 1
    \xintloop
      \toks4 \expandafter{\the\MAT_cntd}%
      \expandafter\def\expandafter\MAT_tmpd\expandafter
        {\csname MAT@\string\MAT_invN{\the\toks0}{\the\toks4}\endcsname }%
      \expandafter\def\expandafter\MAT_tmpe\expandafter
        {\csname MAT@\string\MAT_invN{\the\toks2}{\the\toks4}\endcsname }%
      \expandafter\let\expandafter\MAT_tmpc\MAT_tmpd
      \expandafter\expandafter\expandafter\let\expandafter\MAT_tmpd\MAT_tmpe
      \expandafter\let\MAT_tmpe\MAT_tmpc
    \ifnum \MAT_cntd<\MAT_cntb
        \advance\MAT_cntd \xint_c_i
    \repeat
}%

\def\MATinv_Ipivot {%
% does pivot simplification on both matrices M and N
% pivot is from matrice M at location (cntb,cnta)
    \expandafter\def\expandafter\MAT_tmpa\expandafter {\the\MAT_cnta}%
    \expandafter\def\expandafter\MAT_tmpb\expandafter {\the\MAT_cntb}%
    \expandafter\let\expandafter\MAT_pivot
    \csname MAT@\string\MAT_invM{\the\MAT_cntb}{\the\MAT_cnta}\endcsname 
    \MAT_cntd \MAT_cnta 
    \xintloop
      \ifnum\MAT_cntd<\MAT_cntc
      \advance\MAT_cntd\xint_c_i
    % divide in M all entries to the right of the pivot by pivot 
     \expandafter\def\expandafter\MAT_tmpd\expandafter {\the\MAT_cntd}%
     \expandafter
          \edef\csname MAT@\string\MAT_invM{\MAT_tmpb}{\MAT_tmpd}\endcsname
     {\MAT_DIV{\csname MAT@\string\MAT_invM{\MAT_tmpb}{\MAT_tmpd}\endcsname }
              {\MAT_pivot}}%
    \repeat
    \MAT_cntd \xint_c_i
    \xintloop
    % divide in N all elements on the "b" row with column indices at most
    % equal to "b" by the pivot value
    \edef\MAT_tmpd {\the\MAT_cntd}
     \expandafter
          \edef\csname MAT@\string\MAT_invN{\MAT_tmpb}{\MAT_tmpd}\endcsname
     {\MAT_DIV{\csname MAT@\string\MAT_invN{\MAT_tmpb}{\MAT_tmpd}\endcsname }%
              {\MAT_pivot}%
      }%
    \ifnum\MAT_cntd<\MAT_cntb
    \advance\MAT_cntd \xint_c_i
    \repeat
  % we now will simplify the next rows, in both matrices M and N
  % Again we don't have to do all entries: >a in M and <= b in N
    \MAT_cntd \MAT_cntb % will be increased by 1, row index
    \xintloop
    {% will not create a group!
    \ifnum\MAT_cntd<\MAT_cntc 
    \advance\MAT_cntd \xint_c_i % we start with the "b+1" row
    % We are working with row \cntd
    \edef\MAT_tmpd {\the\MAT_cntd}%
    % we need the (\cntd, \cnta) entry
    \edef\MAT_tmpf 
        {\csname MAT@\string\MAT_invM{\MAT_tmpd}{\MAT_tmpa}\endcsname }%
    % We now multiply by tmpf the cntb row and subtract it from the cntd row
    % this sets to zero the (cntd,cnta) entry:
    % in matrix M, only need to look at columns to the right
    \MAT_cnte\MAT_cnta % necessarily cnta< size of M, as cnta<= cntb<cntd
    \advance\MAT_cnte \xint_c_i 
    \xintloop
        \edef\MAT_tmpe {\the\MAT_cnte}%
        \expandafter
        \edef\csname MAT@\string\MAT_invM{\MAT_tmpd}{\MAT_tmpe}\endcsname 
        {\MAT_SUB{\csname MAT@\string\MAT_invM{\MAT_tmpd}{\MAT_tmpe}\endcsname }
           {\MAT_MUL \MAT_tmpf
            {\csname MAT@\string\MAT_invM{\MAT_tmpb}{\MAT_tmpe}\endcsname }}%
         }%
    \ifnum\MAT_cnte<\MAT_cntc
        \advance\MAT_cnte \xint_c_i
    \repeat% end of subloop for matrix M, row "d", columns "e>=a"
    % we now do the row "d" in matrix N, columns "e<=b"
    \MAT_cnte \xint_c_i
    \xintloop
        \edef\MAT_tmpe {\the\MAT_cnte}%
        \expandafter
        \edef\csname MAT@\string\MAT_invN{\MAT_tmpd}{\MAT_tmpe}\endcsname 
        {\MAT_SUB{\csname MAT@\string\MAT_invN{\MAT_tmpd}{\MAT_tmpe}\endcsname }
           {\MAT_MUL \MAT_tmpf
            {\csname MAT@\string\MAT_invN{\MAT_tmpb}{\MAT_tmpe}\endcsname }}%
         }%
    \ifnum\MAT_cnte<\MAT_cntb
      \advance\MAT_cnte \xint_c_i
    \repeat% end of subloop for matrix N, row "d"
   }\repeat 
}% 

\def\MATinv_IIpivot {%
% does pivot simplification on matrices M and N
% M is now upper triangular with 1's on the diagonal
% pivot = 1 is in the \MAT_cnta row. We simplify rows above.
% There is no need to keep track of the computations for M itself
% Only need to read M data and modify rows of N accordingly
    \expandafter\def\expandafter\MAT_tmpa\expandafter {\the\MAT_cnta}%
    \MAT_cntb \MAT_cnta
    \xintloop
    {% will not create a group!
    \ifnum\MAT_cntb>\xint_c_i 
       \advance\MAT_cntb -\xint_c_i 
    \expandafter\def\expandafter\MAT_tmpb\expandafter {\the\MAT_cntb}%
    \expandafter\let\expandafter\MAT_tmpf 
        \csname MAT@\string\MAT_invM{\MAT_tmpb}{\MAT_tmpa}\endcsname
    \MAT_cntd\xint_c_i 
    \xintloop
        \expandafter\def\expandafter\MAT_tmpd\expandafter {\the\MAT_cntd}%
        \expandafter
        \edef\csname MAT@\string\MAT_invN{\MAT_tmpb}{\MAT_tmpd}\endcsname 
        {\MAT_SUB
           {\csname MAT@\string\MAT_invN{\MAT_tmpb}{\MAT_tmpd}\endcsname }
           {\MAT_MUL \MAT_tmpf
            {\csname MAT@\string\MAT_invN{\MAT_tmpa}{\MAT_tmpd}\endcsname }}%
         }%
    \ifnum\MAT_cntd<\MAT_cntc
       \advance\MAT_cntd \xint_c_i
    \repeat
   }\repeat
}% 


% DISPLAYING MACROS
\makeatother

\def\MATraw {\MATrawwith {\MATrawone}}%

\def\MATrawone {\xintPRaw}%

\def\MATrawwith #1#2{%
     \xintListWithSep {; }%
     {\xintApply { \MAT_raw_row {#1}#2}{\xintSeq {1}{#2[I]}}}%
}%
\def\MAT_raw_row #1#2#3{%
    \xintListWithSep {, }%
    {\xintApply { \MAT_raw_one {#1}#2{#3}}{\xintSeq {1}{#2[J]}}}%
}%
\def\MAT_raw_one #1#2#3#4{#1{#2[#3,#4]}}%

%% MATH MODE DISPLAYING

\newcommand\MATdisplay [1][1.25]{\MATdisplaywith [#1]{\MATdisplayone}}

\def\MATdisplayone {\xintSignedFrac}

\newcolumntype\MATdisplaycoltype {c}
\newcolumntype\MATdisplaypreamble [1]{@{}*{#1[J]}\MATdisplaycoltype@{}}

\newcommand\MATdisplaywith [3][1.25]
   {\left(\def\arraystretch{#1}%
    \begin{array}{\MATdisplaypreamble {#3}}
         \xintListWithSep {\\}
       {\xintApply { \MAT_display_row {#2}#3}{\xintSeq {1}{#3[I]}}}
    \end{array}\right)%
}%

\def\MAT_display_row #1#2#3{%
    \xintListWithSep {&}
     {\xintApply{ \MAT_display_one {#1}#2{#3}}{\xintSeq {1}{#2[J]}}}%
}%

\def\MAT_display_one #1#2#3#4{#1{#2[#3,#4]}}%

\def\MATminus     {\expandafter\MAT_minus_a\romannumeral-`0}%
\def\MAT_minus_a  {\futurelet\XINT_token\MAT_minus_b }%
\def\MAT_minus_b  {\ifx\XINT_token-\else\phantom{-}\fi }%

\usepackage {siunitx}
\usepackage {numprint}

\newcommand{\MATfloatdisplay}[1][\XINTdigits]
           {\MATfloatdisplaywith [#1]{\MATfloatone}}%

\def\MATfloatone #1{\expandafter\MAT_flone\romannumeral-`0#1!}%

\def\MAT_flone #1.#2e#3!{%
    \xintSgnFork{\xintiiSgn{\XINT_Abs #3}}%
    {}{#1.#2}{#1.#2\times 10^{#3}}}%

\newcommand{\MATfloatdisplaywith}[3][\XINTdigits]
   {\left(\edef\MAT_tmpa{#1}%
    \begin{array}{\MATdisplaypreamble{#3}}
     \xintListWithSep {\\}
       {\xintApply { \MAT_fldisplay_row {#2}#3}{\xintSeq {1}{#3[I]}}}%
    \end{array}\right)}%

\def\MAT_fldisplay_row #1#2#3{%
    \xintListWithSep {&}
     {\xintApply{ \MAT_fldisplay_one {#1}#2{#3}}{\xintSeq {1}{#2[J]}}}}%

\def\MAT_fldisplay_one #1#2#3#4{#1{\xintFloat [\MAT_tmpa]{#2[#3,#4]}}}%

\catcode`_ 8   

\begin{document}
see the images.
\end{document}

逆

矩阵 矩阵 矩阵iii 矩阵 矩阵v

答案4

您还可以使用sagetex包裹，使用免费软件Sage。

优点：

  • 可维护性
  • 全部功能Sage：矩阵，多项式，绘图等......以及任何类型的操作（例如编辑中所需的操作！）
  • 不要重新发明轮子，造一辆自行车吧！
  • 轻松导出至或集成至 LaTeX
  • 如果需要的话可以轻松包含源代码
  • 免费软件！

缺点：

  • 需要您的计算机或服务器上的 Sage 来执行计算
  • 需要 LaTeX 之外的一些编译

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