Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 27/refcontrol

Exercises

Exercise Create referencenumber

Determine the eigenvectorsMDLD/eigenvectors of the function ${}\mathbb {R} \rightarrow \mathbb {R} ,x\mapsto \pi x$.

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Check whether the vector ${}{\begin{pmatrix}3\\1\\-1\end{pmatrix}}$ is an eigenvectorMDLD/eigenvector for the matrix

${\begin{pmatrix}-2&-5&1\\0&-2&2\\4&-3&5\end{pmatrix}}.$

In this case, determine the corresponding eigenvalue.MDLD/eigenvalue

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Determine the eigenvectorsMDLD/eigenvectors and the eigenvaluesMDLD/eigenvalues for a linear mappingMDLD/linear mapping

$\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2},$

given by a matrix of the form ${}{\begin{pmatrix}a&b\\0&d\end{pmatrix}}$.

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Show that the first standard vectorMDLD/standard vector is an eigenvector for every upper triangular matrix.MDLD/upper triangular matrix What is its eigenvalue?MDLD/eigenvalue

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Let

${}M={\begin{pmatrix}d_{1}&\ast &\cdots &\cdots &\ast \\0&d_{2}&\ast &\cdots &\ast \\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1}&\ast \\0&\cdots &\cdots &0&d_{n}\end{pmatrix}}\,$

be an upper triangular matrix.MDLD/upper triangular matrix Show that an eigenvalueMDLD/eigenvalue of ${}M$ is a diagonal entry of ${}M$.

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Determine the eigenvalues, eigenvectors and eigenspaces for a plane rotationMDLD/plane rotation ${}{\begin{pmatrix}\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}$, with rotation angle ${}\alpha$, ${}0\leq \alpha <2\pi$, over ${}\mathbb {R}$.

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Show that every matrixMDLD/matrix ${}M\in \operatorname {Mat} _{2}(\mathbb {C} )$ has at least one eigenvalue.MDLD/eigenvalue

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Let

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

be endomorphismsMDLD/endomorphisms on a ${}K$-vector spaceMDLD/vector space ${}V$, and let ${}v\in V$ be an eigenvectorMDLD/eigenvector of ${}\varphi$ and of ${}\psi$. Show that ${}v$ is also an eigenvector of ${}\varphi \circ \psi$. What is its eigenvalue?

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Let ${}\varphi \colon V\rightarrow V$ be an isomorphismMDLD/isomorphism (vs) on a ${}K$-vector spaceMDLD/vector space ${}V$, and let ${}\varphi ^{-1}$ be its inverse mapping.MDLD/inverse mapping Show that ${}a\in K$ is an eigenvalueMDLD/eigenvalue of ${}\varphi$ if and only if ${}a^{-1}$ is an eigenvalue of ${}\varphi ^{-1}$.

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Give an example of a linear mappingMDLD/linear mapping

$\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$

such that ${}\varphi$ has no eigenvalue,MDLD/eigenvalue but a certain powerMDLD/power (endomorphism) ${}\varphi ^{n}$, ${}n\geq 2$, has an eigenvalue.

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Let ${}K$ be a field, and let ${}\varphi \colon V\rightarrow V$ be an endomorphismMDLD/endomorphism (linear) on a ${}K$-vector spaceMDLD/vector space ${}V$, satisfying

${}\varphi ^{n}=\operatorname {Id} _{V}\,$

for a certain ${}n\in \mathbb {N}$. Show that every eigenvalueMDLD/eigenvalue ${}\lambda$ of ${}\varphi$ fulfills the property ${}\lambda ^{n}=1$.

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Let ${}K$ be a field, and let ${}\varphi \colon V\rightarrow V$ be an endomorphismMDLD/endomorphism (linear) on a ${}K$-vector spaceMDLD/vector space ${}V$. Let ${}\lambda \in K$ be an eigenvalueMDLD/eigenvalue of ${}\varphi$ and ${}P\in K[X]$ a polynomial.MDLD/polynomial (1K) Show that ${}P(\lambda )$ is an eigenvalue of ${}P(\varphi )$.

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Let ${}M$ be a square matrix, which can be written as a block matrix

${}M={\begin{pmatrix}A&0\\0&B\end{pmatrix}}\,$

with square matrices ${}A$ and ${}B$. Show that a number ${}\lambda \in K$ is an eigenvalueMDLD/eigenvalue of ${}M$ if and only if ${}\lambda$ is an eigenvalue of ${}A$ or of ${}B$.

===Exercise Exercise 27.14

change===

Let ${}K$ be a field,MDLD/field ${}V$ a ${}K$-vector spaceMDLD/vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping.MDLD/linear mapping Show that the following statements hold.

1. Every eigenspaceMDLD/eigenspace

   $\operatorname {Eig} _{\lambda }{\left(\varphi \right)}$

   is a linear subspaceMDLD/linear subspace of ${}V$.

2. ${}\lambda$ is an eigenvalueMDLD/eigenvalue for ${}\varphi$ if and only if the eigenspace ${}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}$ is not the null space.MDLD/null space
3. A vector ${}v\in V,\,v\neq 0$, is an eigenvectorMDLD/eigenvector for ${}\lambda$ if and only if ${}v\in \operatorname {Eig} _{\lambda }{\left(\varphi \right)}$.

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Let ${}V=\mathbb {R} [X]_{\leq d}$ denote the set of all real polynomials of degree ${}\leq d$. Determine the eigenvalues, eigenvectors and eigenspaces of the derivation operator

$V\longrightarrow V,P\longmapsto P'.$

The concept of an eigenvector is also defined for vector spaces of infinite dimension, the relevance can be seen already in the following exercise.

Exercise Create referencenumber

Let ${}V$ denote the real vector space that consists of all functions from ${}\mathbb {R}$ to ${}\mathbb {R}$ that are arbitrarily often differentiable.

a) Show that the derivation ${}f\mapsto f'$ is a linear mappingMDLD/linear mapping from ${}V$ to ${}V$.

b) Determine the eigenvaluesMDLD/eigenvalues of the derivation and determine, for each eigenvalue, at least one eigenvector.MDLD/eigenvector

c) Determine for every real number the eigenspaceMDLD/eigenspace and its dimension.MDLD/dimension (vs)

===Exercise Exercise 27.17

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Let ${}K$ be a field,MDLD/field ${}V$ a ${}K$-vector spaceMDLD/vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping.MDLD/linear mapping Show that

${}\operatorname {ker} {\left(\varphi \right)}=\operatorname {Eig} _{0}{\left(\varphi \right)}\,.$

Exercise Create referencenumber

Let ${}K$ be a field, and let ${}\varphi \colon V\rightarrow V$ be an endomorphismMDLD/endomorphism (linear) on a ${}K$-vector spaceMDLD/vector space ${}V$. Let ${}\lambda \in K$ and let

${}U=\operatorname {Eig} _{\lambda }{\left(\varphi \right)}\,$

be the corresponding eigenspace.MDLD/eigenspace Show that ${}\varphi$ can be restricted to a linear mapping

$\varphi {|}_{U}\colon U\longrightarrow U,v\longmapsto \varphi (v),$

and that this mapping is the homothetyMDLD/homothety with scale factor ${}\lambda$.

===Exercise Exercise 27.19

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Let ${}K$ be a field,MDLD/field ${}V$ a ${}K$-vector spaceMDLD/vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping.MDLD/linear mapping Let ${}\lambda _{1}\neq \lambda _{2}$ be elements in ${}K$. Show that

${}\operatorname {Eig} _{\lambda _{1}}{\left(\varphi \right)}\cap \operatorname {Eig} _{\lambda _{2}}{\left(\varphi \right)}=0\,.$

===Exercise Exercise 27.20

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Let ${}K$ be a field,MDLD/field ${}V$ a finite-dimensionalMDLD/finite-dimensional (vs) ${}K$-vector spaceMDLD/vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping.MDLD/linear mapping Show that there exist at most ${}\dim _{K}{\left(V\right)}$ many eigenvaluesMDLD/eigenvalues for ${}\varphi$.

Hand-in-exercises

Exercise (1 mark) Create referencenumber

Check whether the vector ${}{\begin{pmatrix}6\\-5\\8\end{pmatrix}}$ is an eigenvectorMDLD/eigenvector of the matrix

${\begin{pmatrix}-8&0&-1\\2&4&-5\\-3&7&2\end{pmatrix}}.$

In this case, determine the corresponding eigenvalue.MDLD/eigenvalue

Exercise (1 mark) Create referencenumber

Check whether the vector ${}{\begin{pmatrix}7\\2\\3\end{pmatrix}}$ is an eigenvectorMDLD/eigenvector of the matrix

${\begin{pmatrix}2&-4&-9\\-2&7&-2\\-1&-1&0\end{pmatrix}}.$

In this case, determine the corresponding eigenvalue.MDLD/eigenvalue

Exercise (4 (1+3) marks) Create referencenumber

The nightlife in the village Kleineisenstein consists in the following three opportunities: to stay in bed (at home), the pub "Nightowl“ and the dancing club "Pirouette“. In a night, one can observe within an hour the following movements:

a) ${}1/10$ of the people in bed go to the Nightowl, ${}1/12$ go to the Pirouette and the rest stays in bed.

b) ${}1/3$ of the people in the Nightowl go to the Pirouette, ${}1/5$ go to bed and the rest stays in the Nightowl.

c) ${}3/5$ of the people in the Pirouette stay in the Pirouette, a percentage of ${}8$ go to the Nightowl, the rest goes to bed.

1. Establish a matrix, which describes the movements within an hour.
2. Kleineisenstein has ${}500$ inhabitants. Determine a distribution of the inhabitants (on the three locations), which does not change within an hour.

Exercise (3 marks) Create referencenumber

Let ${}K$ be a field, and let ${}\varphi \colon V\rightarrow V$ be an endomorphismMDLD/endomorphism (linear) on a ${}K$-vector spaceMDLD/vector space ${}V$. Show that ${}\varphi$ is a homothetyMDLD/homothety if and only if every vector ${}v\in V$, ${}v\neq 0$, is an eigenvectorMDLD/eigenvector of ${}\varphi$.

Exercise (4 marks) Create referencenumber

Consider the matrix

${}M={\begin{pmatrix}1&1\\-1&1\end{pmatrix}}\,.$

Show that ${}M$, as a real matrix, has no eigenvalue.MDLD/eigenvalue Determine the eigenvalues and the eigenspacesMDLD/eigenspaces of ${}M$ as a complexMDLD/complex (number) matrix.

Exercise (6 marks) Create referencenumber

Consider the realMDLD/real matricesMDLD/matrices

${}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \operatorname {Mat} _{2}(\mathbb {R} )\,.$

Characterize, in dependence on ${}a,b,c,d$, when such a matrix has

1. two different eigenvalues,MDLD/eigenvalues
2. one eigenvalue with a two-dimensional eigenspace,MDLD/eigenspace
3. one eigenvalue with a one-dimensional eigenspace,MDLD/eigenspace
4. no eigenvalue.

Footnotes

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