Set mapping/Pointed/Corresponding linear mapping/Exercise

We consider, on the set

{}S=\{1,\ldots ,n,*\}\,,

the set of mappings

{}B={\left\{\pi :S\rightarrow S\mid \pi (*)=*\right\}}\,.

For ${}\pi \in B$ , we assign (for a fixed field ${}K$ ) the linear mapping

\varphi _{\pi }\colon K^{n}\longrightarrow K^{n}

given by

{}\varphi _{\pi }(e_{i})={\begin{cases}e_{\pi (i)},{\text{ if }}\pi (i)\neq *\,,\\0,{\text{ if }}\pi (i)=*\,.\end{cases}}\,

We denote by ${}M_{\pi }$ the corresponding matrix with respect to the standard basis.

a) Establish the matrix ${}M_{\pi }$ , in case ${}n=4$ , for the following ${}\pi$ :

(1)

${}x$	${}1$	${}2$	${}3$	${}4$	${}*$
${}\pi (x)$	${}2$	${}*$	${}3$	${}*$	${}*$

(2)

${}x$	${}1$	${}2$	${}3$	${}4$	${}*$
${}\pi (x)$	${}*$	${}1$	${}2$	${}3$	${}*$

(3)

${}x$	${}1$	${}2$	${}3$	${}4$	${}*$
${}\pi (x)$	${}*$	${}*$	${}*$	${}*$	${}*$

(4)

${}x$	${}1$	${}2$	${}3$	${}4$	${}*$
${}\pi (x)$	${}2$	${}2$	${}2$	${}2$	${}*$

b) What properties hold for the columns and for the rows of ${}M_{\pi }$ ?

c) For what ${}\pi$ is ${}M_{\pi }$ bijective?

d) For what ${}\pi$ is ${}M_{\pi }$ nilpotent?

e) What is the dimension of the kernel of ${}M_{\pi }$ ?

f) Show

{}M_{\pi \circ \rho }=M_{\pi }\circ M_{\rho }\,.

g) Show that every nilpotent ${}n\times n$ -matrix ${}M$ is similar to a matrix of the form ${}M_{\pi }$ .