Passing an argument with the alignment operator (&) to a macro within another aligned environment

Question

I am trying to write macros for quickly formatting matrices in math mode. Here is a simple, working example:

\documentclass{article}
\usepackage{amsmath}
\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}

\begin{document}
\begin{displaymath}
M = \pmat{1&0\\0&1}
\end{displaymath}
\end{document}

This also works if used within an aligned environment (a table, align environment, cases environment, etc.):

\documentclass{article}
\usepackage{amsmath}
\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}

\begin{document}
\begin{align}
M &= \pmat{1&0\\0&1}
\end{align}
\end{document}

Normally, within the align environment the & symbol tells LaTeX to start a new column, but since the matrix values are contained between {} group tokens, LaTeX does not attempt to parse the 1&0\\0&1 argument and thus everything works as it was intended.

Let's create a new macro that does not use the group tokens {} to denote the macro argument. For example, it is fairly common to use square brackets to group an argument as in this working example (try it!):

\documentclass{article}
\usepackage{amsmath}
\newcommand{\bmat}[1][]{\begin{bmatrix}#1\end{bmatrix}}

\begin{document}
\begin{displaymath}
M = \bmat[1&0\\0&1]
\end{displaymath}
\end{document}

So far, so good. Here comes the problem! If I try to use my new macro \bmat within another aligned environment:

% DOES NOT WORK!!!
\documentclass{article}
\usepackage{amsmath}
\newcommand{\bmat}[1][]{\begin{bmatrix}#1\end{bmatrix}}

\begin{document}
\begin{align}
M &= \bmat[1&0\\0&1]
\end{align}
\end{document}

I get an error because LaTeX tries to interpret the & within the matrix argument as part of the outer align environment.

The difference between the way LaTeX treats the first macro \pmat{} and the second one \bmat[] is that the group tokens {} prevent LaTeX from parsing the argument contained within the curly braces until after the macro \pmat has been called. On the other hand, when LaTeX sees the \bmat[arg] macro, the argument is not surrounded by group tokens and so LaTeX tries to parse it before getting eaten up by \bmat. The extra & character screws up the syntax of the align environment and so LaTeX crashes. At least, that is my understanding of what is going on, and I could be wrong!

So, the question is whether or not it is possible to force LaTeX to treat the argument in the [] case just as it did in the {} case (without causing additional problems). In other words, I want to be able to write a macro a-la \bmat[1&0\\0&1] that works in aligned environments but retains the square bracket syntax.

CONCLUSION:

Here is the final macro with working [] syntax, thanks to egreg (requires xparse and must be contained in a separate package file or between \makeatletter and \makeatother in preamble):

\DeclareDocumentCommand\mat{}{{\ifnum\z@=`}\fi\@mat}
\DeclareDocumentCommand\@mat{ g o d() d|| }
{
    \IfNoValueTF{#1}
    {
        \IfNoValueTF{#2}
        {
            \IfNoValueTF{#3}
            {
                \IfNoValueTF{#4}
                {()}
                {\begin{vmatrix}#4\end{vmatrix}}
            }
            {
                \begin{pmatrix}#3\end{pmatrix}
                \IfNoValueTF{#4}{}{|#4|}
            }
        }
        {
            \begin{bmatrix}#2\end{bmatrix}
            \IfNoValueTF{#3}{}{(#3)}
            \IfNoValueTF{#4}{}{|#4|}
        }
    }
    {
        \begin{matrix}#1\end{matrix}
        \IfNoValueTF{#2}{}{[#2]}
        \IfNoValueTF{#3}{}{(#3)}
        \IfNoValueTF{#4}{}{|#4|}
    }
    \ifnum\z@=`{\fi}
}

This macro works like an overloaded function with different argument delimiters {}, [], (), or ||. There are probably many reasons why this is a terrible idea --- so please, post your comments below. I would like to hear from those who know better than me.

Answer 1

You can do it, but I strongly recommend you not doing it. You gain nothing in using [ and ] as delimiters for a mandatory argument.

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newcommand\bmat{{\ifnum\z@=`}\fi\@bmat}
\def\@bmat[#1]{\begin{bmatrix}#1\end{bmatrix}\ifnum\z@=`{\fi}}
\makeatother

\begin{document}
$\bmat[1&0\\0&1]$
\begin{align}
M &= \bmat[1&0\\0&1]
\end{align}
\end{document}

(Note: I have reduced the text width to shorten the output.)

When you understand the trick, you are allowed to use it. :) Hint: look for \eegroup in Appendix D of the TeXbook.

---

Defining \mat to look for the next token in order to decide a delimiter is, in my opinion, wrong. I'd do like this:

\newcommand{\mat}[2][]{\begin{#1matrix}#1\end{#1matrix}}

so that

\mat{1&2\\3&4}    % no fences
\mat[b]{1&2\\3&4} % brackets
\mat[p]{1&2\\3&4} % parentheses
\mat[v]{1&2\\3&4} % vertical lines
\mat[V]{1&2\\3&4} % double vertical lines
\mat[B]{1&2\\3&4} % braces

would do even better, with minimum hassle and better readability.

Personally, I would stick with the long form, but it's a matter of taste.

Answer 2

Why this happens is rather interesting, since of course the raw code

\begin{align}
 M = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\end{align}

does work even though the array separators are not between braces. The reason is that bmatrix itself inserts braces specifically to make this work, and by the time scanning has reached the separators, TeX has already read the opening brace and knows that it should protect them.

In your code, it would seem that \bmat is replaced with its expansion, which is the above code, before the & is read, but in fact, the & has already been read at that point: it needs to be picked up by the macro argument grabber. That is to say, even though the & wasn't "executed" it was seen, and apparently, TeX will detect the end of an array cell even just when going through tokenization.

Because of this, there is no way to use the normal \newcommand\bmat[1][]{} syntax to define a command that "just works" in arrays; egreg's answer shows how you can roll your own that accepts the same syntax but sneakily inserts braces. It's probably also possible to make a \catcode change prior to scanning for the argument that would make [] into functional brace characters, but for all that effort, I suggest you redesign your macro not to use an optional argument instead. As a perhaps compromise, you can place braces around it: {\bmat[1&0\\0&1]}, as Scott H. suggested.