Vectors and Matrices in Trace Diagrams
How matrices operate when inserted into trace diagrams… includes lots of examples.

Source Code
$$ABC \leftrightarrow
\tikz[heighttwo]{\draw(0,0)to(0,2)node[small matrix,ellipse,pos=.5]{$ABC$};}
=\tikz[heighttwo]{\draw(0,0)to(0,2)
node[small matrix,pos=.25]{$C$}node[small matrix,pos=.5]{$B$}node[small matrix,pos=.75]{$A$};}
\quad\text{and}\quad
\bfv^T A\bfw \leftrightarrow
\tikz[heightoneonehalf]{
\draw(0,0)node[vector]{$\bfw$}to[with small matrix={$A$}](0,1.5)node[vector]{$\bfv$};},$$
Basic Vectors

%vector
$\tikz{\draw(0,.2)node[vector]{$v$}to(0,.8);}$
%covector
$\tikz{\draw(0,.2)to(0,.8)node[vector]{$v$};}$
Matrix Coefficients
The matrix coefficients or elements:

$$a_{ij}=\bse^i A\bse_j \leftrightarrow
\tikz[heightoneonehalf]{\draw(0,0)node[vector]{$i$}to[with small matrix={$A$}](0,1.5)node[vector]{$j$};}.$$
Trivalent Diagrams

\tikz[trivalent,heighttwo]{
\draw(0,0)node[small vector]{${\sf n}_{n-l}$}
to node[small matrix]{$\mathbf{X}$}node[pos=.75,rightlabel]{$n$}(0,2)node[small vector]{${\sf n}_{n-k}$};}
Trace, Determinant, Pfaffian, and Other Invariants
See this snippet.
page revision: 8, last edited: 15 Feb 2010 15:40