Perform matrix operations (addition, product, transpose, etc.) in LaTeX?

Question

I know the calc package can perform infix-notation arithmetic in LaTeX... but I want more!

I'd like to perform (not necessarily infix-notation) linear-algebra operations such as scalar multiplication, matrix addition and product in LaTeX, and then print the result in an array or in some matrix environment from amsmath.

Why would I want to do that in LaTeX directly? Why do I not simply use some linear-algebra software, such as Matlab, Mathematica, etc.?

Well, suppose I want to walk my readers through a detailed linear-algebra calculation with many numerical examples. Of course, I could perform all the steps manually first and then hardcode the result of each step in my input file. However, this approach

• is prone to errors (LaTeX' arrays are not very user-friendly to typeset),
• lacks maintainability (I may decide to change the data in my example, which means I have to modify everything that follows).

Hence my question: Is there a way of easily performing linear-algebra operations in LaTeX?

---

Ideally, I would like to

1. mimick Matlab's syntax for defining matrices (using commas as column separators and semicolons as end-of-row characters), performing operations on them, extracting sub matrices, etc.. The syntax could be something like the following:

   \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. have a command that typesets a "matrix object" in an array or matrix environment, e.g.

   \typesetmatrix[bmatrix]{A}
   

3. be able to perform operations on matrices of arbitrary--albeit relatively small--dimensions (edit: not just 2x2 and 3x3 as in the calculator package).

Is that currently possible with some package? If not, I'm considering rolling up my sleeves and implementing something, but this would probably prove quite difficult, and I would like to avoid reinventing the wheel :)

---

Edit about other operations that would be useful:

• diag (extraction of the diagonal of a square matrix)
• trace
• determinant
• norm(s)
• condition number(s)
• inverse

Even more advanced matrix operations that would be awesome, but probably tough to implement:

• eigenvalues & eigenvectors,
• QR, least squares etc.
• SVD,
• other common matrix factorisations.

Answer 1

Here is some code to manipulate matrices of any size. Currently, it can perform additions, subtractions, and multiplication (as well as fetching individual entries, and transposing a matrix, for instance). Entries are floating points that l3fp supports (16 digits of precision).

% 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}

Edit: added \matget, which extracts an individual entry in the matrix.

Answer 2

calculator package might help.

Answer 3

Matrix operations (muliplications, inverses, determinants) either exactly or with floats (arbitrary precision)

this answer from November 2013 was edited in March 2017 because already in late 2014, removal of a macro from xint had rendered code not usable, and since 1.1 (2014/10/28) xintfrac does not load xinttools automatically.

\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}

---

Answer 4

This is one more option you can check out. Asymptote supports matrix operations, and here is a brief example to demonstrate what is possible. It includes matrix expressions, transpose and inverse. Usage:

• define matrices inside the asy environment along with operations on them;

• define TeX names with matrixdata function, e.g.: matrixdata("D^T",transpose(a*(b-a)));, here a TeX name for typesetting a matrix is D^T, and the matrix is a result of the matrix expression transpose(a*(b-a)), where a,b were previously defined.

• access matrix data inside a standard matrix environment with \mxData{} , e.g. \mxData{D^T}

Example matr.tex:

\documentclass[10pt,a4paper]{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage[inline]{asymptote}
\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}

\begin{asydef}
typedef real[][] matrix;

string smatrixdata(string texName, matrix a){
  string s="\expandafter\gdef\csname texMatrix["+texName+"]\endcsname{";
  for(int i=0;i<a.length;++i){
    for(int j=0;j<a[0].length;++j){
      s+=((j==0)?"":"&")+format("%#.3f",a[i][j]);
    }
    if(i<a.length-1){
      s+="\\"+'\n';
    }
  }
  s+="}";
  return s;
}

void matrixdata(string texName, matrix a){
  tex(smatrixdata(texName, a));
};

\end{asydef}

\gdef\mxData#1{\ifcsname texMatrix[#1]\endcsname\relax\csname texMatrix[#1]\endcsname\relax\fi}


\begin{document}

\begin{asy}
  matrix a={
  {-8,-1,6,0},
  {10,-4,-5,-5},
  {-2,-5,-8,2},
  {-4,7,9,-3},
  };

  matrix b={
  {3,-3,9,-9},
  {-5,9,6,7},
  {-9,-8,-6,1},
  {7,-4,-9,9},
  }; 

  matrix a_squared;
  a_squared=a*a;

  matrixdata("A",a);
  matrixdata("B",b);
  matrixdata("C",a*b);
  matrixdata("A^2",a_squared);
  matrixdata("D",a*(b-a));

  matrixdata("D^T",transpose(a*(b-a)));

  matrix va={ 
    {10},
    {20}
  };

  matrix vb={ 
    {1,2,3,4,5,6}
  };

  matrixdata("va",va);
  matrixdata("vb",vb);
  matrixdata("va*vb",va*vb);
  matrixdata("A^-1",inverse(a));
  matrixdata("A*A^-1",a*inverse(a));

\end{asy}


\begin{align}
A&=\left[
\begin{matrix}
\mxData{A}
\end{matrix}
\right]
\\
B&=\left[
\begin{matrix}
\mxData{B}
\end{matrix}
\right]
\\
C=A\times B&=\left[
\begin{matrix}
\mxData{C}
\end{matrix}
\right]
\\
A^2&=\left[
\begin{matrix}
\mxData{A^2}
\end{matrix}
\right]
\\
D=A\times (B-A)&=\left[
\begin{matrix}
\mxData{D}
\end{matrix}
\right]
\\
D^T&=\left[
\begin{matrix}
\mxData{D^T}
\end{matrix}
\right]
\\
a&=\left[
\begin{matrix}
\mxData{va}
\end{matrix}
\right]
\\
b&=\left[
\begin{matrix}
\mxData{vb}
\end{matrix}
\right]
\\
a\times b&=\left[
\begin{matrix}
\mxData{va*vb}
\end{matrix}
\right]
\\
A^{-1}&=\left[
\begin{matrix}
\mxData{A^-1}
\end{matrix}
\right]
%
\\
A\times A^{-1}&=\left[
\begin{matrix}
\mxData{A*A^-1}
\end{matrix}
\right]
%
\end{align}

\end{document}

To process it with latexmk, create file latexmkrc:

sub asy {return system("asy '$_[0]'");}
add_cus_dep("asy","eps",0,"asy");
add_cus_dep("asy","pdf",0,"asy");
add_cus_dep("asy","tex",0,"asy");

and run latexmk -pdf matr.tex.

As for the other operations you mentioned, feel free to add them as a functions inside the asydef block. I suppose, the C-implementations of the algorithms could be found somewhere, and since the Asymptote syntax is very similar, a translation should not be difficult.

Answer 5

You could also use the sagetex package, working with the free software Sage.

Pros:

• Maintainability
• Full power of Sage: matrices, but also polynomials, plots, etc... and any kind of operations (such as the ones required in the edit!)
• Don't reinvent the wheel, build a bike!
• Easy export to — or integration into — LaTeX
• Easy inclusion of the source code if this is needed
• Free software!

Cons:

• Needs Sage on your computer, or a server to perform computations
• Needs some compilation outside LaTeX