%Works well with \documentclass[12pt]{amsart}

%: ****************
%: description of \numericalmatrix
% Make it easy to write matrices with all items flush right.
% First argument optional - gives outer symbol   and  if it's missing you get bracket 
%                                              bracket           gives left/right bracket pair            [ ]
%                                              brace              gives left/right brace pair               { }
%                                              parentheses gives left/right parentheses pair   ( )
%                                              line                 gives left/right vertical line             | |
%                                              a space         gives no left/right symbols
% second and third argument are numbers m and n for an m\times n matrix (rows times columns)
% The last entry is the elements in the matrix, across and then down.
% Remark - The individual matrix entries are gobbled by the TeX input mechanism so if an
%                   entry consists of more than one symbol you will need to { - } it.
% Remark - if you usually want ()'s instead of []'s just change the [bracket] to what you want
% Remark - You can also get pretty much what you want: [\left\{,\right]] will give a left brace 
%                   and a right bracket.
% Remark - If you precede your entry in the [ ] by a +, the new value becomes global until
%                  changed again.
% Remark - If the last entry in the matrix is complicated you may need an extra set of {}'s
%                   to avoid the last entry being split apart and the second part discarded. MAYBE FIXED
%
% There are two commands \entry and \CR. \entry has one variable and handles the display
% of each matrix entry. The default value is \hfill #1 which gives right justification.
% \CR by default is \cr and ends each matrix line. If you need a \noalign... at the end of
% each line, just redefine \CR
% When a given matrix entry is outputted by \entry, there are two global variables set,
% \cR and \cC (current Row and current Column) which are set to these values and can be
% used to adjust the display. For example
%     \renewcommand{\entry}[1]{\hfill \ifnum\cR=\cC{#1}_{\textcolor{red}{\the\cR \the\cC}}\else%
%     \ifnum\cR<\cC{#1}_{\textcolor{green}{\the\cR \the\cC}}\else%
%     {#1}_{\textcolor{blue}{\the\cR \the\cC}}\fi\fi}
% typesets the i j-th entry and adds a subscript i j which is red for diagonal elements, blue below
% the diagonal and green above.

% There is a global list, \matrixlist which has the matrix encoded so to print it all one needs to
% do is $\begin{matrix}\matrixlist\end{matrix}$ to print it a second time

% The last used left deliminator is in \leftb and the right one is in \rightb

% uniqueness string nbx
% Means all the internal routines and variables have an nbx in them.
%:  \numericalmatrix

% The last argument of \numericalmatrix and \NM is optional. If it is present, it consists
% two dimensions separated by a comma. One or both can be missing. Examples are
% [2pt, 3pt] [,5pt] [6pt,]
% Missing dimensions are supplied by a default value. If either of the dimensions is preceded
% by a +, that becomes the new default value

% Matrices can be nested EXCEPT that auto from \rigidmatrix does not work.

\def\rightb{}\gdef\rightmatrixdeliminatorskipSave{}\gdef\rightmatrixdeliminatorskip{}%

\newcommand{\numericalmatrix}[4][\defaultmatrixdeliminator]{%
\edef\rightBsave{\rightb}%
\edef\rightmatrixdeliminatorskipSave{\rightmatrixdeliminatorskip}%
\nbxsetBraces{#1}%
\nbxsetMatrixList{#2}{#3}{#4}%
\gdef\nbxA{#2}\gdef\nbxB{#3}\gdef\nbxC{#4}%
\NMnbxcollectskips%
}

%:  \NM 
% short form - uses \defaultmatrixdeliminator
\newcommand{\NM}[3]{
\numericalmatrix[\defaultmatrixdeliminator]{#1}{#2}{#3}%
}

\newcount\cCSave
%:       \nbxNM
% most of the work is here
\newcommand{\nbxNM}[3]{%
\def\nbxll##1{%  set the macro that sets the matrix entries
\nxbEntry{\hfill##1\hfill}%
\global\advance\cC by 1 %
\ifnum\cC>\maxCLRT \global\cC=1 \fi%
\ifnum\cC=1 \relax\nxbCR\else&\fi}%
%
{\cCSave=\cC %
\cC=1 % initialize \cC 
\edef\maxRLRT{\maxR}\edef\maxCLRT{\maxC}%
\showMatrix%
%
\gdef\nxbEntry{\defaultentry}%
\gdef\nxbCR{\defaultCR}%
\global\cC = \cCSave }%
}

%:       \showMatrix
\newcommand{\showMatrix}[1][\matrixlist]{%
\leftb\hskip\leftmatrixdeliminatorskip\begin{matrix}
#1%
\end{matrix}\hskip\rightmatrixdeliminatorskip\rightb%
}
%:        \NMnbxcollectskips
\newcommand{\NMnbxcollectskips}[1][,]{\def\nbxD{#1}%
\expandafter\nbxlrmd\nbxD @%
\nbxNM{\nbxA}{\nbxB}{\nbxC}%
\xdef\rightb{\rightBsave}%
\xdef\rightmatrixdeliminatorskip{\rightmatrixdeliminatorskipSave}%
}

%:  \resetEntry
\def\resetEntry#1{\gdef\nxbEntry{#1}}
\resetEntry{\defaultentry}
%:  \resetCR
\def\resetCR#1{\gdef\nxbCR{#1}}
\resetCR{\defaultCR}

% Outline of the code
% The first subroutine determines the left and right deliminators
% \nbxsetMatrixList is a loop which collects the entries.
% At the end, \matrixlist is a list: { \nbxll{a_11} ... \nbxll{a_1n} ...
% \nbxll{a_m1} ...  \nbxll{a_m n} }.
% \NMnbxcollectskips collects the last optional argument and gets the \hskip's 
% between the deliminators and the entries.
% Then \nbxNM defines \nbxll and executes the list which in the default case puts an \hfill before
% each entry and a & after all of them except the last one in each column for which it
% inserts a \CR after the entry.
% It also sets \cR to the current row and \cC to the current column so you can use
% this data in your \entry macro if you wish.

%%%%%%%%%%%%%%%

%: ****************
%: description of \rigidmatrix - \RM is the short version
% Same as \numericalmatrix with 1 more variable
% #5 gives the fixed column width 
% A value of auto for #5 will cause TeX to calculate all the natural column widths and then use the
% maximum of them for all the column widths.
% If the entry is preceded by a +, the value becomes global (including +auto).
% There is a final, optional pair of dimensions just like \numericalmatrix

%:  \rigidmatrix
\newcommand{\rigidmatrix}[5][\defaultmatrixdeliminator]{%
\edef\rightBsave{\rightb}%
\edef\rightmatrixdeliminatorskipSave{\rightmatrixdeliminatorskip}%
\nbxsetBraces{#1}%
\nbxsetMatrixList{#2}{#3}{#4}%
\nbxsetColumnSkips{#5}%
\gdef\nbxA{#2}\gdef\nbxB{#3}\gdef\nbxC{#4}%
\RMnbxcollectskips%
}

%:  \RM
% short form - uses \defaultmatrixdeliminator and default spacings 
\newcommand{\RM}[3]{%
\rigidmatrix[\defaultmatrixdeliminator]{#1}{#2}{#3}{}%
}

\newdimen\zznbxcolumnwidthSave
%:  \nbxRM
% most of the work is here
\newcommand{\nbxRM}[3]{
%
\gdef\nbxll##1{% set the macro that sets the matrix entries
\hbox to\zznbxcolumnwidth{$\nxbEntry{##1}$}%
\global\advance\cC by 1 %
\ifnum\cC>\maxC \global\cC=1 \fi%
\ifnum\cC=1 \relax\nxbCR\else&\fi}%
%
{\cCSave=\cC %
\zznbxcolumnwidthSave=\zznbxcolumnwidth%
\cC=1 % initialize \cC 
\showMatrix%
\gdef\nxbEntry{\defaultentry}%
\gdef\nxbCR{\defaultCR}%
\global\cC = \cCSave %
\global\zznbxcolumnwidthSave=\zznbxcolumnwidth}%
}

%:  \RMnbxcollectskips
\newcommand{\RMnbxcollectskips}[1][,]{\def\nbxD{#1}%
\expandafter\nbxlrmd\nbxD @%
\nbxRM{\nbxA}{\nbxB}{\nbxC}%
\xdef\rightb{\rightBsave}%
\xdef\rightmatrixdeliminatorskip{\rightmatrixdeliminatorskipSave}%
}

%%%%%%%%%%%%%%%%%%%
%: ****************
%: \setMatrixList 
\newcommand{\setMatrixList}[4]{%
\gdef#4{}%
\gdef\maxR{#1}\gdef\maxC{#2}%
\global\nbxms=\maxR \global\multiply\nbxms by \maxC%
\nbxNRs=#1\relax\nbxNCs=#2\gdef\nbxMEs{#3@}%
\nbxloop\ifnum\nbxms>0\relax\global\advance\nbxms by -1%
\expandafter\nbxnV\nbxMEs{#4}\relax%
\nbxrepeat%
}

%%%%%%%%%%%%%%%%%%%
%: ****************
%: \pushState 
\gdef\pushState{%
\gdef\qnbxpush##1##2{\nbxsaveNumber\nbxstateNumber\of\nbxstateList\macro{##1}\label{##2}}%
\global\let\exnbx=\qnbxpush%
\gnbxmatrixg%
\global\advance\nbxstateNumber by 1 %
}

%: \popState
\gdef\popState{%
\global\advance\nbxstateNumber by -1 %
\gdef\qnbxpop##1##2{\nbxrestore\nbxstateNumber\list{nbxstateList}\label{##2}\to{##1}}%
\global\let\exnbx=\qnbxpop%
\gnbxmatrixg%
}

%: ****************
%: \transpose
\gdef\xxtransposematrix#1\tomatrix#2{\gdef#2{}%
\let\xdf=\nbxll%
%
\def\nbxll##1{%
\showthe\nbxIndex\show#2\showthe\nbxwhichIndex%
\ifnum\nbxIndex=\nbxwhichIndex\relax%
\nbxa={\nbxll{##1}}\nbxb=\expandafter{#2}\xdef#2{\the\nbxb\the\nbxa}\fi%
\advance\nbxIndex by 1}%
%
\nbxXX=0%
\nbxMM=\maxC \multiply\nbxMM by \maxR%
\nbxCC=1\nbxRR=1% \CC + \maxC*\RR -\maxC
\nbxloop%
\ifnum\nbxXX< \nbxMM \advance\nbxXX by 1%
\nbxwhichIndex=-\maxC%
\advance \nbxwhichIndex by \nbxRR%
\nbxTmpB=\nbxCC \multiply\nbxTmpB by\maxC%
\advance \nbxwhichIndex by \nbxTmpB%
\advance\nbxCC by 1 \ifnum\nbxCC>\maxR \nbxCC=1 \advance\nbxRR by 1\fi%
\relax\nbxIndex=1\relax#1%
\nbxrepeat%
\let\nbxll=\xdf%
\edef\nbxMM{\maxC}\xdef\maxC{\maxR}\xdef\maxR{\nbxMM}%
\global\cC=1 \global\cR=1%
}

\gdef\transposematrix#1\tomatrix#2{\gdef#2{}%
\let\xdf=\nbxll%
%
\def\nbxll##1{\relax%
\ifnum\nbxIndex=\nbxwhichIndex\relax%
\nbxa={\nbxll{##1}}\nbxb=\expandafter{#2}\xdef#2{\the\nbxb\the\nbxa}\fi%
\advance\nbxIndex by 1}%
%
\nbxXX=0%
\nbxMM=\maxC \multiply\nbxMM by \maxR%
\nbxCC=1\nbxRR=1% \CC + \maxC*\RR -\maxC
\nbxloop%
\ifnum\nbxXX< \nbxMM \advance\nbxXX by 1%
\nbxwhichIndex=-\maxC%
\advance \nbxwhichIndex by \nbxCC%
\nbxTmpB=\nbxRR \multiply\nbxTmpB by\maxC%
\advance \nbxwhichIndex by \nbxTmpB%
\advance\nbxRR by 1 \ifnum\nbxRR>\maxR \nbxRR=1 \advance\nbxCC by 1\fi%
\relax\nbxIndex=1\relax#1%
\nbxrepeat%
\let\nbxll=\xdf%
\edef\nbxMM{\maxC}\xdef\maxC{\maxR}\xdef\maxR{\nbxMM}%
\global\cC=1 \global\cR=1%
}
%%
%: **************** 
%: subroutines

%:                \nbxlrmd
\gdef\nbxlrmd#1,#2@{%
\def\xxnbx{#1}%
\ifx\xxnbx\empty\gdef\leftmatrixdeliminatorskip{\defaultleftmatrixdeliminatorskip}\else%
\expandafter\nbxPluser#1@{\leftmatrixdeliminatorskip}{\defaultleftmatrixdeliminatorskip}\fi%
\def\xxnbx{#2}%
\ifx\xxnbx\empty\gdef\rightmatrixdeliminatorskip{\defaultrightmatrixdeliminatorskip}\else%
\expandafter\nbxPluser#2@{\rightmatrixdeliminatorskip}{\defaultrightmatrixdeliminatorskip}\fi%
}

%:                \nbxleftright
\def\nbxleftright#1,#2@{\gdef\leftb{#1}\gdef\rightb{#2}}

%:                \nbxsetBraces
\def\nbxsetBraces#1{
\edef\xnbxx{#1}%
\ifx\xnbxx\empty\relax\def\xnbxx{bracket}\else%
\expandafter\nbxPluser#1@{\zzdefaultdeliminator}{\defaultmatrixdeliminator}%
\xdef\xnbxx{\zzdefaultdeliminator}\fi%
\ifx\xnbxx\nbxbr \gdef\leftb{\left[}\gdef\rightb{\right]}\else%
\ifx\xnbxx\nbxbs \gdef\leftb{\left\{}\gdef\rightb{\right\}}\else%
\ifx\xnbxx\nbxpt \gdef\leftb{\left(}\gdef\rightb{\right)}\else%
\ifx\xnbxx\nbxv \gdef\leftb{\left\vert}\gdef\rightb{\right\vert}\else%
\ifx\xnbxx\nbsp \gdef\leftb{}\gdef\rightb{}\else%
\expandafter\nbxleftright\xnbxx @\relax%
\fi\fi\fi\fi\fi%
}

%:                \nbxPluser
\def\nbxPluser#1#2@#3#4{\def\xnbxx{#1}\ifx\xnbxx\nbxPlus\relax\gdef#3{#2}%
\gdef#4{#2}\else\gdef#3{#1#2}\fi}

\def\nbxPlus{+}
\def\nbxaa{auto}
\newdimen\zznbxcolumnwidth
\newbox\nbxmeasurebox

%:                \nbxsetColumnSkips
\newcount\cXC
\def\nbxsetColumnSkips#1{
\def\xnbxx{#1}\ifx\xnbxx\empty\relax\xdef\zznbxcolumnwidthmacro{\nbxcolumnwidthmacro}\else%
\expandafter\nbxPluser#1@{\zznbxcolumnwidthmacro}{\nbxcolumnwidthmacro}\fi%
%
\ifx\zznbxcolumnwidthmacro\nbxaa\relax\global\zznbxcolumnwidth=0pt%
%
\global\cXC=1 %
\edef\lrtmaxCX{\maxC}%
\def\nbxll##1{%
\setbox\nbxmeasurebox =\hbox{$\nxbEntry{##1}$}%
\ifdim\wd\nbxmeasurebox>\zznbxcolumnwidth\relax%
\global\zznbxcolumnwidth=\wd\nbxmeasurebox \fi%
\global\advance\cXC by 1 %
\ifnum\cXC>\lrtmaxCX \global\cXC=1 \fi}%
%
\matrixlist%
\else%
\global\zznbxcolumnwidth=\zznbxcolumnwidthmacro\fi%
}

\gdef\nbxcolumnwidthmacro{0pt}
\gdef\nbxleftsideskip{0pt}
\gdef\nbxrightsideskip{0pt}

\newcount\nbxms

%:                \nbxsetMatrixList
\def\nbxsetMatrixList#1#2#3{%
\setMatrixList{#1}{#2}{#3}{\matrixlist}%
}

%:                \nbxloop
\def\nbxloop#1\nbxrepeat{\def\nbxbody{#1}\nbxiterate}
\def\nbxiterate{\nbxbody\let\nbxnextB=\nbxiterate\else\let\nbxnextB=\relax\fi\nbxnextB}



%:                \nbxnV
\def\nbxnV#1#2@#3{\nbxa={\nbxll{#1}}%
\nbxb=\expandafter{#3}%
\xdef#3{\the\nbxb\the\nbxa}%
\ifnum\nbxms=1 \gdef\nbxMEs{{#2}@}\else\gdef\nbxMEs{#2@}\fi%
}

%%% Like a \pushState and \popState


%:                \nbxsaveNumber
\gdef\nbxsaveNumber#1\of#2\macro#3\label#4{%
\nbxmpC=#1%
\long\def\\##1{\advance\nbxmpC-1 \ifnum\nbxmpC=0 
\global\expandafter\let\csname##1#4\endcsname=#3\fi}#2}
 
%:                \nbxrestore
\gdef\nbxrestore#1\list#2\label#3\to#4{%
\nbxmpC=#1%
\long\def\\##1{\advance\nbxmpC-1 \ifnum\nbxmpC=0 
\global\let#4=##1\fi}\expandafter\csname#2#3\endcsname}

%%%%%%%%%
%: ****************
%: variables, etc.
\newcount\nbxNCs\newcount\nbxNRs\newcount\nbxmpC\newcount\nbxvtc
\newtoks\nbxa\newtoks\nbxb
\newtoks\nbxc\newtoks\nbxd


\gdef\nbxstateList{\\{A}\\{B}\\{C}\\{D}\\{E}\\{F}\\{G}\\{H}\\{I}\\{J}\\{K}\\{L}\\{M}\\{N}}
\gdef\nbxmoose{%
\\{\csname A\the\nbxa\endcsname}%
\\{\csname B\the\nbxa\endcsname}%
\\{\csname C\the\nbxa\endcsname}%
\\{\csname D\the\nbxa\endcsname}%
\\{\csname E\the\nbxa\endcsname}%
\\{\csname F\the\nbxa\endcsname}%
\\{\csname G\the\nbxa\endcsname}%
\\{\csname H\the\nbxa\endcsname}%
\\{\csname I\the\nbxa\endcsname}%
\\{\csname J\the\nbxa\endcsname}%
\\{\csname K\the\nbxa\endcsname}%
\\{\csname L\the\nbxa\endcsname}%
\\{\csname M\the\nbxa\endcsname}%
\\{\csname N\the\nbxa\endcsname}%
}

\gdef\gnbxmatrixg{\exnbx{\defaultentry}{defaultentry}\exnbx{\defaultCR}{defaultCR}%
\exnbx{\maxR}{nbxmaxR}%
\exnbx{\maxC}{nbxmaxC}\exnbx{\leftb}{nbxleftb}\exnbx{\rightb}{nbxrightb}%
\exnbx{\defaultmatrixdeliminator}{nbxdefaultmatrixdeliminator}}

\let\\=\relax
\def\ww#1#2{%
\nbxa={#2}%
\expandafter\xdef\csname nbxstateList\the\nbxa\endcsname{%
\nbxmoose}}

\let\exnbx=\ww
\gnbxmatrixg


\newcount\nbxstateNumber\nbxstateNumber=1
\def\defaultentry#1{\hfill#1}
\newcommand{\defaultCR}{\cr}
\newcount\cR\newcount\cC


\newcommand{\nbxll}[1]{}
\newcommand{\nbxLL}[1]{}

\gdef\defaultleftmatrixdeliminatorskip{0pt}
\gdef\defaultrightmatrixdeliminatorskip{0pt}

\newcommand{\defaultmatrixdeliminator}{}
\def\nbxbr{bracket}
\def\nbxbs{brace}
\def\nbxpt{parentheses}
\def\nbxv{line}
\def\nbsp{blank}


\newcount\nbxIndex\newcount\nbxwhichIndex
\newcount\nbxXX\newcount\nbxCC\newcount\nbxRR\newcount\nbxMM
\newcount\nbxTmpB

%: \vstrut
\newcommand{\vstrut}[2]{%
\vrule height #1pt depth #2pt width 0pt}