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){$A$};}
\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);
}
\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);
}

page revision: 17, last edited: 15 Feb 2010 16:36