Matrix Minors and Cofactors

Snippets Page

Trace diagrams for matrix minors and cofactors.

Source Code

[A_{I,J}]
    =\sgn(J^c\:\overset\leftarrow{J})
    \tikz{
        \node[ciliatednode=180](va)at(0,.2){}
            edge[bend left]node[reverse matrix on edge,pos=.65]{$A$}(-.7,1.2)
            edge[bend left]node[reverse matrix on edge,pos=.65]{$A$}(-.4,1.2)
            edge[bend right]node[matrix on edge,pos=.65]{$A$}(.7,1.2)
            \foreach\xa/\xb in{right/-.7,left/.4,left/.7}{edge[bend \xa](\xb,-.5)};
        \draw(0,1.2)node[toplabel]{$I_1I_2\cdots I_k$};
        \draw(.2,-.5)node[bottomlabel]{$J_1^c\:\cdots\:J_{n-k}^c$};}
    =\sgn(I^c\:\overset\leftarrow{I})
    \tikz{
        \node[ciliatednode=180](va)at(0,.2){}
            edge[bend left]node[matrix on edge,pos=.65]{$A$}(-.7,1.2)
            edge[bend left]node[matrix on edge,pos=.65]{$A$}(-.4,1.2)
            edge[bend right]node[reverse matrix on edge,pos=.65]{$A$}(.7,1.2)
            edge[bend right](-.7,-.5)
            edge[bend left](.4,-.5)
            edge[bend left](.7,-.5);
        \draw(0,1.2)node[toplabel]{$J_1J_2\cdots J_k$};
        \draw(.2,-.5)node[bottomlabel]{$I_1^c\:\cdots\:I_{n-k}^c$};
    }.

Cofactors

    C_{I,J}=
    \frac{\tdsign{n}}{(n-k)!}
    \tikz[heighttwo,scale=1.2]{
        \node[ciliatednode=180](vb)at(0,1.6){}
            edge[bend left](-.7,2.2)
            edge[bend left](-.4,2.2)
            edge[bend right](.7,2.2);
        \node[ciliatednode=180](va)at(0,.3){}
            edge[out=160,in=200,looseness=2]node[matrix on edge]{$A$}(vb)
            edge[out=20,in=-20,looseness=2]node[matrix on edge]{$A$}(vb)
            edge[out=20,in=-20,looseness=1.2]node[matrix on edge]{$A$}(vb)
            edge[bend right](-.7,-.2)
            edge[bend right](-.4,-.2)
            edge[bend left](.7,-.2);
        \node[bottomlabel]at(0,-.2){$J_1J_2\cdots J_k$};
        \node[toplabel]at(0,2.2){$I_1I_2\cdots I_k$};
        \draw[dotdotdot](-.6,1)to node[toplabel,scale=.8]{$n\!-\!k$}(.4,1);
    }

The Adjugate

\mathsf{adj}(A)
    =\frac{\tdsign{n}}{(n-1)!}
    \tikz[heighttwo,scale=1.2]{
        \node[ciliatednode=160](vb)at(0,1.6){}edge(0,2.2);
        \node[ciliatednode=200](va)at(0,.3){}
            edge[out=160,in=200,looseness=1.7]node[reverse matrix on edge]{$A$}(vb)
            edge[out=20,in=-20,looseness=1.6]node[reverse matrix on edge]{$A$}(vb)
            edge[out=20,in=-20,looseness=1.1]node[reverse matrix on edge]{$A$}(vb)
            edge(0,-.2);
        \draw[dotdotdot](-.4,.9)to node[toplabel,scale=.7]{$n\!-\!1$}(.2,.9);
    }

Invariant Nodes

\tikz[scale=1.5,shift={(0,.25)}]{
        \node[ciliatednode=180](va)at(0,.5){}
            edge[bend right]node[matrix on edge,pos=.7]{$A$}(-.7,-.2)
            edge[bend right]node[matrix on edge,pos=.7]{$A$}(-.4,-.2)
            edge[bend left]node[reverse matrix on edge,pos=.7]{$A$}(.7,-.2);
        \draw[dotdotdot](-.4,.1)to node[bottomlabel]{$n$}(.6,.1);
        \draw[dotdotdot](-.4,.1)to(.6,.1);
    }=\det(A)
    \tikz[scale=1.5,shift={(0,.25)}]{
        \node[ciliatednode=180](va)at(0,.5){}
            edge[bend right](-.7,-.2)edge[bend right](-.4,-.2)edge[bend left](.7,-.2);
        \draw[dotdotdot](-.4,.1)to node[bottomlabel]{$n$}(.6,.1);
        \draw[dotdotdot](-.4,.1)to(.6,.1);
    }

\tikz[scale=1.3]{
        \node[ciliatednode=180](va)at(0,.5){}
            edge[bend left](-.7,1.2)
            edge[bend left](-.4,1.2)
            edge[bend right](.7,1.2)
            edge[bend right]node[matrix on edge,pos=.7]{$A$}(-.7,-.5)
            edge[bend left]node[reverse matrix on edge,pos=.7]{$A$}(.4,-.5)
            edge[bend left]node[reverse matrix on edge,pos=.7]{$A$}(.7,-.5);
        \draw[dotdotdot](-.4,.9)to node[toplabel]{$k$}(.6,.9);
        \draw[dotdotdot](-.6,-.4)to node[bottomlabel,scale=.8]{$n\!-\!k$}(.4,-.4);
    }=\det(A)
    \tikz[scale=1.3]{
        \node[ciliatednode=180](va)at(0,.2){}
            edge[bend left]node[matrix on edge,pos=.65]{$\overline{A}$}(-.7,1.2)
            edge[bend left]node[matrix on edge,pos=.65]{$\overline{A}$}(-.4,1.2)
            edge[bend right]node[reverse matrix on edge,pos=.65]{$\overline{A}$}(.7,1.2)
            edge[bend right](-.7,-.5)
            edge[bend left](.4,-.5)
            edge[bend left](.7,-.5);
        \draw[dotdotdot](-.6,-.2)to node[bottomlabel,scale=.8]{$n\!-\!k$}(.4,-.2);
        \draw[dotdotdot](-.4,.9)to node[toplabel]{$k$}(.6,.9);
    }