Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 27

Exercise for the break

Exercise

Show that the matrix

${\begin{pmatrix}-2&4\\-1&2\end{pmatrix}}$

is nilpotent.

Exercises

Exercise

Show that the complex matrix

${\begin{pmatrix}1&{\mathrm {i} }\\{\mathrm {i} }&-1\end{pmatrix}}$

is nilpotent.

Exercise

We consider the matrix

${}M={\begin{pmatrix}0&a&b&c&d\\0&0&e&f&g\\0&0&0&h&i\\0&0&0&0&j\\0&0&0&0&0\end{pmatrix}}\,$

over a field ${}K$. Show that the fifth power of ${}M$ equals ${}0$, that is,

${}M^{5}=MMMMM=0\,.$

Exercise

Let ${}A={\left(a_{ij}\right)}$ and ${}B={\left(b_{ij}\right)}$ be square matrices of length ${}n$. Suppose that ${}a_{ij}=0$ holds for ${}j\leq i+d$, and ${}b_{ij}=0$ holds for ${}j\leq i+e$ for some ${}d,e\in \mathbb {Z}$. Show that the entries ${}c_{ij}$ of the product ${}AB$ fulfill the condition ${}c_{ij}=0$ for ${}j\leq i+d+e+1$.

Exercise

Let

$D\colon \mathbb {R} [X]_{\geq m}\longrightarrow \mathbb {R} [X]_{\geq m}$

be the restriction of the derivation operator ${}P\mapsto P'$ to the space of polynomials of degree ${}\leq m$. Show that ${}D$ is nilpotent. Show also that

$D\colon \mathbb {R} [X]\longrightarrow \mathbb {R} [X]$

is not nilpotent.

Exercise

Let ${}V$ be a ${}K$-vector space, and

$\varphi \colon V\longrightarrow V$

an injective linear mapping. Show that ${}\varphi$ is not nilpotent.

Exercise

Let ${}K$ be a field, and let ${}V$ denote a finite-dimensional ${}K$-vector space. Let

$\varphi \colon V\longrightarrow V$

be a nilpotent linear mapping. Show that ${}\varphi ^{n}=0$, where ${}n$ is the dimension of ${}V$.

Exercise

Let ${}K$ be a field, and let ${}V$ denote a ${}K$-vector space. Let

$\varphi \colon V\longrightarrow V$

be a nilpotent linear mapping. Show that ${}0$ is the only eigenvalue of ${}\varphi$.

Exercise

Let ${}K$ be a field, and let ${}V$ denote a finite-dimensional ${}K$-vector space. Let

$\varphi \colon V\longrightarrow V$

be a nilpotent linear mapping. Suppose that ${}\varphi$ is diagonalizable. Show

${}\varphi =0\,.$

Exercise

Let ${}K$ be a field, and let ${}V$ denote a finite-dimensional ${}K$-vector space. Let

$\varphi \colon V\longrightarrow V$

be a nilpotent linear mapping. What is the determinant of ${}\varphi$?

Exercise

Let ${}K$ be a field, and let ${}V$ denote a finite-dimensional ${}K$-vector space. Let

$\varphi \colon V\longrightarrow V$

be a nilpotent linear mapping. What is the trace of ${}\varphi$?

Exercise

Let ${}K$ be a field.

a) Characterize the nilpotent ${}2\times 2$-matrices

${\begin{pmatrix}x&y\\z&w\end{pmatrix}}$

over ${}K$ by two equations in the variables ${}x,y,z,w$.

b) Are these equations linear?

Exercise

a) Let ${}M$ be a ${}2\times 2$-matrix that is trigonalizable, but neither diagonalizable nor invertible. Show that ${}M$ is nilpotent.

b) Give an example of a ${}3\times 3$-matrix ${}M$ that is trigonalizable, but neither diagonalizable, nor invertible, nor nilpotent.

Exercise

Give an example of a linear mapping ${}\varphi \colon K^{3}\rightarrow K^{3}$ that is trigonalizable, where the trace and the determinant are ${}0$, and that is not nilpotent.

Exercise

Show that the linear subspaces ${}U_{i}$, constructed in the proof of Lemma 27.11 , are not ${}\varphi$-invariant in general.

Exercise

Let ${}K$ be a field, and let ${}V$ denote a finite-dimensional ${}K$-vector space. Let

$\varphi \colon V\longrightarrow V$

be a nilpotent linear mapping. Suppose that the kernel of ${}\varphi$ is one-dimensional. Let

${}V_{i}=\operatorname {kern} \varphi ^{i}\,,$

and let ${}s$ be the minimal number with

${}\varphi ^{s}=0\,.$

1. Show that all ${}V_{i}$, ${}1\leq i\leq s$, have a direct decomposition

   ${}V_{i}=V_{i-1}\oplus U_{i}\,,$

   where ${}U_{i}$ is one-dimensional.

2. Show that the restrictions

   $\varphi \colon U_{i}\longrightarrow V_{i-1}$

   are bijective for ${}1<i<s$.

3. Show that ${}s$ equals the dimension of ${}V$.

Exercise

Let

$\varphi \colon V\longrightarrow V$

be a nilpotent endomorphism on a finite-dimensional ${}K$-vector space ${}V$. Let

${}V_{i}:=\operatorname {kern} \varphi ^{i}\,.$

Show that for the dimension jumps, the relation

${}\dim _{K}{\left(V_{i}\right)}-\dim _{K}{\left(V_{i}-1\right)}\geq \dim _{K}{\left(V_{i+1}\right)}-\dim _{K}{\left(V_{i}\right)}\,$

holds.

Exercise

Show that there exists a family of (up to) ${}2^{n-1}$ ${}n\times n$-matrices with the property that every nilpotent endomorphism on an ${}n$-dimensional vector space ${}V$, can be described, in a suitable basis, by one of these matrices.

Exercise

Show that every nilpotent endomorphism, on a four-dimensional space, can be brought into exactly one of the following forms.

${\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}},\,{\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{pmatrix}},\,{\begin{pmatrix}0&1&0&0\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{pmatrix}},\,{\begin{pmatrix}0&0&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0\end{pmatrix}},\,{\begin{pmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\0&0&0&0\end{pmatrix}}.$

Exercise

Let ${}v_{1},\ldots ,v_{6}$ be a basis of the ${}K$-vector space ${}V$, and let ${}\varphi \colon V\rightarrow V$ denote the linear mapping, given by

$v_{1},v_{2}\mapsto v_{4},\,v_{3},v_{5}\mapsto v_{6},\,v_{4}\mapsto 0,\,v_{6}\mapsto v_{2}.$

1. Show that ${}\varphi$ is nilpotent.
2. Determine the minimal ${}s$ satisfying ${}\varphi ^{s}=0$.
3. Determine the kernel of ${}\varphi$.
4. Find a basis of ${}V$ such that the describing matrix of ${}\varphi$, with respect to this basis, is in Jordan normal form.

The following exercise generalizes the idea to assign a permutation matrix to a permutation.

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 ${}M_{\pi }$.

Exercise

Let ${}V$ be a ${}K$-vector space, and let

$\varphi \colon V\longrightarrow V$

denote a nilpotent linear mapping. Let

$\psi \colon V\longrightarrow V$

be another linear mapping satisfying

${}\psi \circ \varphi =\varphi \circ \psi \,.$

Show that ${}\psi \circ \varphi$ is also nilpotent.

Exercise *

Give an example of two nilpotent linear mappings

$\varphi ,\psi \colon K^{2}\longrightarrow K^{2}$

such that neither ${}\varphi \circ \psi$ nor ${}\varphi +\psi$ are nilpotent.

Exercise

Let ${}a\in \mathbb {R}$ be a real number with ${}\vert {a}\vert <1$. As is known from analysis, we have

${}\lim _{n\rightarrow \infty }a^{n}=0\,.$

Is the linear mapping

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto ax$

nilpotent?

Hand-in-exercises

Exercise (3 marks)

Let ${}M$ be a ${}2\times 2$-matrix over a field ${}K$. Show that ${}M$ is nilpotent if and only if the determinant and the trace of ${}M$ are ${}0$.

Exercise (2 marks)

Let ${}\varphi \colon V\rightarrow V$ be a linear mapping, and let ${}V=U\oplus W$ be a direct sum of ${}\varphi$-invariant linear subspaces. Show that ${}\varphi$ is nilpotent if and only if ${}\varphi {|}_{U}$ and ${}\varphi {|}_{W}$ are nilpotent.

Exercise (5 (1+1+1+2) marks)

Let ${}v_{1},\ldots ,v_{7}$ be a basis of the ${}\mathbb {R}$-vector space ${}V$, and let ${}\varphi \colon V\rightarrow V$ denote the linear mapping given by

$v_{3},v_{2}\mapsto v_{5},\,v_{5}\mapsto 4v_{6},\,v_{1},v_{4}\mapsto 0,\,v_{6}\mapsto 5v_{4},\,v_{7}\mapsto 3v_{2}.$

1. Show that ${}\varphi$ is nilpotent.
2. Determine the minimal ${}s$ satisfying ${}\varphi ^{s}=0$.
3. Determine den kernel of ${}\varphi$.
4. Find a basis of ${}V$, such that the matrix describing ${}\varphi$ with respect to this basis is in Jordan normal form.

Exercise (3 marks)

Let ${}V$ be a ${}K$-vector space with a basis ${}v_{n}$, ${}n\in \mathbb {N} _{+}$. Let

$\varphi \colon V\longrightarrow V$

be the linear mapping given by

${}\varphi (v_{1})=0\,$

and

${}\varphi (v_{n})=v_{n-1}\,$

for all ${}n\geq 2$. Is ${}\varphi$ nilpotent?

Exercise (4 marks)

Let ${}V$ be a finite-dimensional ${}K$-vector space, and let

$\varphi \colon V\longrightarrow V$

be nilpotent. Show that

${}\psi :=\operatorname {Id} +\varphi \,$

is bijective.

Exercise (4 marks)

Let ${}V$ be a ${}K$-vector space, and let

$\varphi ,\psi \colon V\longrightarrow V$

denote nilpotent linear mappings, satisfying the relation

${}\varphi \circ \psi =\psi \circ \varphi \,.$

Show that also ${}\psi +\varphi$ is nilpotent.

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