labeling diagrams with matrices

Question

I it possible to label an arrow with a matrix in Xy-pic? In the following diagram I want to label the arrows with 2x2 matrices to represent linear maps between vector spaces, in the place of a,b and c.

\documentclass{article}

\usepackage{amsmath}
\usepackage{xypic}

\begin{document}

\begin{equation*}
\xymatrix{
{\bullet}^{V_1}\ar[r]^a &{\bullet}^{V_2}\ar[d]^b
\\&{\bullet}^{V_3}\ar[ul]^c
}
\end{equation*}
\end{document}

It is for a diagram of a quiver representation.

Answer 1

You could just substitute the a, b, c for the matrix or whatever you want in brackets (I've added the - before so that the label is aligned in the middle of the arrow, not the middle of the row/column). Here's an example:

\begin{equation*} 
    \xymatrix @R=2cm @C=2cm { 
    {\bullet}^{V_1} \ar[r]^-{a = \begin{pmatrix}
        a_1 & a_2 \\ a_3 & a_4
    \end{pmatrix}} & {\bullet}^{V_2} \ar[d]^-{b = \begin{pmatrix}
        b_1 & b_2 \\ b_3 & b_4
    \end{pmatrix}} \\
    & {\bullet}^{V_3} \ar[ul]^-{c = \begin{pmatrix}
        c_1 & c_2 \\ c_3 & c_4
    \end{pmatrix}} 
    }
\end{equation*}

And may be you would like to write into an smallmatrix, which could be enough for your needs:

\begin{equation*} 
    \xymatrix @R=2cm @C=2cm { 
    {\bullet}^{V_1} \ar[r]^-{a = \left(\begin{smallmatrix}
        a_1 & a_2 \\ a_3 & a_4
    \end{smallmatrix}\right)} & {\bullet}^{V_2} \ar[d]^-{b = \left(\begin{smallmatrix}
        b_1 & b_2 \\ b_3 & b_4
    \end{smallmatrix}\right)} \\
    & {\bullet}^{V_3} \ar[ul]^-{c = \left(\begin{smallmatrix}
        c_1 & c_2 \\ c_3 & c_4
    \end{smallmatrix}\right)} 
    }
\end{equation*}

The @R=X and the @C=Y in the xymatrix command gives you the chance to change row and column separation.

There is a package tikz-cd (here is the manual at CTAN) which, in my opinion, gives better quality and the language used is easier to understand (i.e. \arrow[hookrightarrow]{rr}{a} instead of the xymatrix \ar@{^{(}->}[rr]^-{a}; or \rar{a} instead of \ar[r]^-{a}). But, in this concrete case there is a problem: tikz-cd causes problems if you want to label an arrow with a matrix (here is a better explanation). That's why I don't include an example here.