Trace, Determinant, and Other Primitive Matrix Invariants

Snippets Page

Diagrams for the trace, the determinant, and other primitive matrix invariants.

Source Code

\chi\left(\tikz{\draw(0,.5)circle(.5);\node[small matrix,yscale=.8]at(.5,.5){$A$};}\right)
    =a_{11}+\cdots+a_{nn}=\tr(A)

Generic Invariants

The family of invariants:

        \tr(A) \propto
        \tikz{\draw(0,.5)circle(.5);\node[small matrix,yscale=.8]at(.5,.5){$A$};}
        \qquad\cdots\qquad
        \tr_i(A) \propto        
    \tikz[heightoneonehalf]{
            \node[ciliatednode=170](topnode)at(0,1.5){};
            \node[ciliatednode=-170](bottomnode)at(0,0){};
            \draw(bottomnode)to[out=180,in=180,looseness=2.3]node[small matrix]{$A$}(topnode);
            \draw(bottomnode)to[out=120,in=-120,looseness=.3]node[small matrix]{$A$}(topnode);
            \draw(bottomnode)to[out=30,in=-30,looseness=.5](topnode);
            \draw(bottomnode)to[out=0,in=0,looseness=2](topnode);
            \draw[dotdotdot](-.9,.75)to node[toplabel]{$i$}(-.3,.75);
            \draw[dotdotdot](.3,.75)to(.9,.75);
        }
        \qquad\cdots\qquad
        \det(A) \propto
        \tikz[heightoneonehalf]{
            \node[ciliatednode=170](topnode)at(0,1.5){};
            \node[ciliatednode=-170](bottomnode)at(0,0){};
            \draw(bottomnode)to[bend left](-.7,.75)node[small matrix]{$A$}to[bend left](topnode);
            \draw(bottomnode)to[bend left](-.4,.75)node[small matrix]{$A$}to[bend left](topnode);
            \draw(bottomnode)to[bend right](.7,.75)node[small matrix]{$A$}to[bend right](topnode);
            \draw[dotdotdot](-.3,.75)to node[toplabel]{$n$}(.6,.75);
        }

The Trace

The trace in terms of matrix elements:

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

The Determinant

General expression:

\tikz[heightoneonehalf]{
            \node[ciliatednode=170](topnode)at(0,1.5){};
            \node[ciliatednode=-170](bottomnode)at(0,0){};
            \draw(bottomnode)to[bend left](-.7,.75)node[small matrix]{$A$}to[bend left](topnode);
            \draw(bottomnode)to[bend left](-.4,.75)node[small matrix]{$A$}to[bend left](topnode);
            \draw(bottomnode)to[bend right](.7,.75)node[small matrix]{$A$}to[bend right](topnode);
            \draw[dotdotdot](-.3,.75)to node[toplabel]{$n$}(.6,.75);
        }
    =(-1)^{\lfloor\frac{n}{2}\rfloor} n! \det(A)

Expression in terms of vectors:

\tikz[scale=1.5,shift={(0,.2)}]{
        \node[ciliatednode=160](va)at(0,.5){}
            edge[bend right]node[small vector,pos=1]{$\bfu_1$}(-.7,-.2)
            edge[bend right]node[small vector,pos=1]{$\bfu_2$}(-.4,-.2)
            edge[bend left]node[small vector,pos=1]{$\bfu_n$}(.7,-.2);
        \draw[dotdotdot](-.4,.1)to node[bottomlabel]{$n$}(.6,.1);
    }
    = \det(\bfu_{1} \cdots \bfu_{n})

Special case for 3d vectors:

\tikz[scale=.6]{
        \node[vertex](node)at(1,1.5){};
        \draw[](0,0)node[vector]{$\bf u$}to[out=90,in=-135](node);
        \draw[](1,0)node[vector]{$\bf v$}to[out=90,in=-90](node);
        \draw[](2,0)node[vector]{$\bf w$}to[out=90,in=-45](node);
    }=\det[\bfu\:\bfv\:\bfw]

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[plain vector]{$\xb$}to node[matrix on edge,pos=.65]{$A$}(\xa,1)
                to[wavyup](\xa,2)node[plain vector]{$\xb$};
        }
        \foreach\xa in {.25,1.75}{\draw[dotdotdot](2,\xa)to(5,\xa);}
        \draw[permutation](.7,1.1)rectangle node[symlabel]{$\sigma$}(5.3,1.5);
    }

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

The Pfaffian

Here is the Pfaffian:

\mathrm{Pf}(A)=
   \tikz[heighttwo]{
        \node[ciliatednode=200]at(0,0){};
        \foreach\xa in{2.5,1.2,.7}{
            \draw(0,0)to[out=180,in=180,looseness=1.7]node[matrix on edge]{$A$}(0,\xa)to[out=0,in=0,looseness=1.7](0,0);
        }
        \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);
   }