Trace, Determinant, and Other Primitive Matrix Invariants

Snippets Page

Diagrams for the trace and determinant.

Source Code

$$
        \tr(A)=
        \tikz{\draw[with small matrix={$A$}](0,.5)circle(.5);}
        \qquad
        \tr_i(A)=C
        \tikz[heightoneonehalf]{
            \node[ciliatednode=140](topnode)at(0,1.5){};
            \node[ciliatednode=220](bottomnode)at(0,0){};
            \foreach\xa/\xb in{90/2.2,30/.5}{
                \draw[with small matrix={$A$},pos=.7](bottomnode)to[bend left=\xa,looseness=\xb](topnode);}
            \foreach\xa/\xb in{90/2,30/.5}{
                \draw(bottomnode)to[bend right=\xa,looseness=\xb](topnode);}
            \foreach\xa/\xb in{-.9/-.3,.2/.9}{\draw[dotdotdot](\xa,.75)to(\xb,.75);}
        }
        \qquad
        \det(A)=C
        \tikz[heightoneonehalf]{
            \node[ciliatednode=140](topnode)at(0,1.5){};
            \node[ciliatednode=220](bottomnode)at(0,0){};
            \draw[with small matrix={$A$}](bottomnode)arc(270:90:.75);
            \draw[with small matrix={$A$}](bottomnode)arc(-90:90:.75);
            \draw[with small matrix={$A$}](bottomnode)to[out=135,in=-135](topnode);
            \draw[dotdotdot](-.25,.75)--(.65,.75);
        }
    $$

Additional Diagrams

The Pfaffian

Here is the Pfaffian:

$$\mathrm{Pf}(A)=
   \tikz[heighttwo]{
        \node[vertex]at(0,0){};
        \foreach\xa in{1.25,.6,.35}{\draw[with small matrix={$A$}](0,\xa)circle(\xa);}
        \draw[dotdotdot](-.5,1)to(-1.1,1.6);\draw[dotdotdot](.5,1)to(1.1,1.6);\draw[dotdotdot](0,1.2)to(0,2.5);
   }
   $$

Trace and Determinant Sum Formulas

The trace in terms of matrix elements:

$$\tr(A) \leftrightarrow \sum_{i=1}^n 
    \tikz[heightoneonehalf]{\draw(0,0)node[vector]{$i$}to[with small matrix={$A$}](0,1.5)node[vector]{$i$};}.$$

The determinant in terms of matrix elements:

$$\det(A)\leftrightarrow \sum_{\sigma\in S_n} \sgn(\sigma)
    \tikz[heighttwo,xscale=.5]{
        \foreach\xa/\xb in{1/1,2/2,5/n}{
            \draw(\xa,0)node[vector]{$\xb$}to[]node[small matrix,pos=.6]{$A$}(\xa,1)
                to[wavyup](\xa,2)node[vector]{$\xb$};
        }
        \foreach\xa in {.25,1.75}{\draw[dotdotdot](2,\xa)to(5,\xa);}
        \draw[antisymmetrizer](.7,1)rectangle node{$\sigma$}(5.3,1.4);
    }
    $$

And a special case of the determinant:

$$\det(A) \leftrightarrow
    \tikz[heightoneonehalf,xscale=.5]{
        \foreach\xa/\xb in{1/1,2/2}{
            \draw(\xa,0)node[vector]{$\xa$}to[]node[small matrix,pos=.8]{$A$}(\xa,.9)
                to[wavyup](\xb,1.5)node[vector]{$\xb$};
        }}
    -
    \tikz[heightoneonehalf,xscale=.5]{
        \foreach\xa/\xb in{1/2,2/1}{
            \draw(\xa,0)node[vector]{$\xa$}to[]node[small matrix,pos=1]{$A$}(\xa,.6)
                to[wavyup](\xb,1.5)node[vector]{$\xb$};
        }}
    $$