Vectors and Matrices in Trace Diagrams

Snippets Page

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.