text: # Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 27
  
  Exercises
  
  ### Exercise
  
  Determine the eigenvectors of the function ${\displaystyle {}\mathbb {R} \rightarrow \mathbb {R} ,x\mapsto \pi x}$.
  
  ### Exercise
  
  Check whether the vector ${\displaystyle {}{\begin{pmatrix}3\\1\\-1\end{pmatrix}}}$ is an eigenvector for the matrix
  
  ${\displaystyle {\begin{pmatrix}-2&-5&1\\0&-2&2\\4&-3&5\end{pmatrix}}.}$
  
  In this case, determine the corresponding eigenvalue.
  
  ### Exercise
  
  Determine the eigenvectors and the eigenvalues for a linear mapping
  
  ${\displaystyle \varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2},}$
  
  given by a matrix of the form ${\displaystyle {}{\begin{pmatrix}a&b\\0&d\end{pmatrix}}}$.
  
  ### Exercise
  
  Show that the first standard vector is an eigenvector for every upper triangular matrix. What is its eigenvalue?
  
  ### Exercise
  
  Let
  
  ${\displaystyle {}M={\begin{pmatrix}d_{1}&\ast &\cdots &\cdots &\ast \\0&d_{2}&\ast &\cdots &\ast \\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1}&\ast \\0&\cdots &\cdots &0&d_{n}\end{pmatrix}}\,}$
  
  be an upper triangular matrix. Show that an eigenvalue of ${\displaystyle {}M}$ is a diagonal entry of ${\displaystyle {}M}$.
  
  ### Exercise
  
  Determine the eigenvalues, eigenvectors and eigenspaces for a plane rotation ${\displaystyle {}{\begin{pmatrix}\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}}$, with rotation angle ${\displaystyle {}\alpha }$, ${\displaystyle {}0\leq \alpha <2\pi }$, over ${\displaystyle {}\mathbb {R} }$.
  
  ### Exercise
  
  Show that every matrix ${\displaystyle {}M\in \operatorname {Mat} _{2}(\mathbb {C} )}$ has at least one eigenvalue.
  
  ### Exercise
  
  Let
  
  ${\displaystyle \varphi ,\psi \colon V\longrightarrow V}$
  
  be an endomorphisms on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$, and let ${\displaystyle {}v\in V}$ be an eigenvector of ${\displaystyle {}\varphi }$ and of ${\displaystyle {}\psi }$. Show that ${\displaystyle {}v}$ is also an eigenvector of ${\displaystyle {}\varphi \circ \psi }$. What is its eigenvalue?
  
  ### Exercise
  
  Let ${\displaystyle {}\varphi \colon V\rightarrow V}$ be an isomorphism on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$, and let ${\displaystyle {}\varphi ^{-1}}$ be its inverse mapping. Show that ${\displaystyle {}a\in K}$ is an eigenvalue of ${\displaystyle {}\varphi }$ if and only if ${\displaystyle {}a^{-1}}$ is an eigenvalue of ${\displaystyle {}\varphi ^{-1}}$.
  
  ### Exercise
  
  Give an example of a linear mapping
  
  ${\displaystyle \varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}}$
  
  such that ${\displaystyle {}\varphi }$ has no eigenvalue, but a certain power ${\displaystyle {}\varphi ^{n}}$, ${\displaystyle {}n\geq 2}$, has an eigenvalue.
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}\varphi \colon V\rightarrow V}$ be an endomorphism on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$, satisfying
  
  ${\displaystyle {}\varphi ^{n}=\operatorname {Id} _{V}\,}$
  
  for a certain ${\displaystyle {}n\in \mathbb {N} }$.[1] Show that every eigenvalue ${\displaystyle {}\lambda }$ of ${\displaystyle {}\varphi }$ fulfills the property ${\displaystyle {}\lambda ^{n}=1}$.
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}\varphi \colon V\rightarrow V}$ be an endomorphism on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$. Let ${\displaystyle {}\lambda \in K}$ be an eigenvalue of ${\displaystyle {}\varphi }$ and ${\displaystyle {}P\in K[X]}$ a polynomial. Show that ${\displaystyle {}P(\lambda )}$ is an eigenvalue[2] of ${\displaystyle {}P(\varphi )}$.
  
  ### Exercise
  
  Let ${\displaystyle {}M}$ be a square matrix, which can be written as a block matrix
  
  ${\displaystyle {}M={\begin{pmatrix}A&0\\0&B\end{pmatrix}}\,}$
  
  with square matrices ${\displaystyle {}A}$ and ${\displaystyle {}B}$. Show that a number ${\displaystyle {}\lambda \in K}$ is an eigenvalue of ${\displaystyle {}M}$ if and only if ${\displaystyle {}\lambda }$ is an eigenvalue of ${\displaystyle {}A}$ or of ${\displaystyle {}B}$.
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, ${\displaystyle {}V}$ a ${\displaystyle {}K}$-vector space and
  
  ${\displaystyle \varphi \colon V\longrightarrow V}$
  
  a linear mapping. Show that the following statements hold.
  
  1. Every eigenspace
  ${\displaystyle \operatorname {Eig} _{\lambda }{\left(\varphi \right)}}$
  
  is a linear subspace of ${\displaystyle {}V}$.
  
  2. ${\displaystyle {}\lambda }$ is an eigenvalue for ${\displaystyle {}\varphi }$ if and only if the eigenspace ${\displaystyle {}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}}$ is not the nullspace.
  3. A vector ${\displaystyle {}v\in V,\,v\neq 0}$, is an eigenvector for ${\displaystyle {}\lambda }$ if and only if ${\displaystyle {}v\in \operatorname {Eig} _{\lambda }{\left(\varphi \right)}}$.
  
  ### Exercise
  
  Let ${\displaystyle {}V=\mathbb {R} [X]_{\leq d}}$ denote the set of all real polynomials of degree ${\displaystyle {}\leq d}$. Determine the eigenvalues, eigenvectors and eigenspaces of the derivation operator
  
  ${\displaystyle V\longrightarrow V,P\longmapsto P'.}$
  
  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
  
  Let ${\displaystyle {}V}$ denote the real vector space, which consists of all functions from ${\displaystyle {}\mathbb {R} }$ tp ${\displaystyle {}\mathbb {R} }$, which are arbitrarily often differentiable.
  
  a) Show that the derivation ${\displaystyle {}f\mapsto f'}$ is a linear mapping from ${\displaystyle {}V}$ to ${\displaystyle {}V}$.
  
  b) Determine the eigenvalues of the derivation and determine, for each eigenvalue, at least one eigenvector.[3]
  
  c) Determine for every real number the eigenspace and its dimension.
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, ${\displaystyle {}V}$ a ${\displaystyle {}K}$-vector space and
  
  ${\displaystyle \varphi \colon V\longrightarrow V}$
  
  a linear mapping. Show that
  
  ${\displaystyle {}\operatorname {ker} {\left(\varphi \right)}=\operatorname {Eig} _{0}{\left(\varphi \right)}\,.}$
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}\varphi \colon V\rightarrow V}$ be an endomorphism on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$. Let ${\displaystyle {}\lambda \in K}$ and let
  
  ${\displaystyle {}U=\operatorname {Eig} _{\lambda }{\left(\varphi \right)}\,}$
  
  be the corresponding eigenspace. Show that ${\displaystyle {}\varphi }$ can be restricted to a linear mapping
  
  ${\displaystyle \varphi {|}_{U}\colon U\longrightarrow U,v\longmapsto \varphi (v),}$
  
  and that this mapping is the homothety with scale factor ${\displaystyle {}\lambda }$.
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, ${\displaystyle {}V}$ a ${\displaystyle {}K}$-vector space and
  
  ${\displaystyle \varphi \colon V\longrightarrow V}$
  
  a linear mapping. Let ${\displaystyle {}\lambda _{1}\neq \lambda _{2}}$ be elements in ${\displaystyle {}K}$. Show that
  
  ${\displaystyle {}\operatorname {Eig} _{\lambda _{1}}{\left(\varphi \right)}\cap \operatorname {Eig} _{\lambda _{2}}{\left(\varphi \right)}=0\,.}$
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, ${\displaystyle {}V}$ a finite-dimensional ${\displaystyle {}K}$-vector space and
  
  ${\displaystyle \varphi \colon V\longrightarrow V}$
  
  a linear mapping. Show that there exist at most ${\displaystyle {}\dim _{}{\left(V\right)}}$ many eigenvalues for ${\displaystyle {}\varphi }$.
  
  Hand-in-exercises
  
  ### Exercise (1 mark)
  
  Check whether the vector ${\displaystyle {}{\begin{pmatrix}6\\-5\\8\end{pmatrix}}}$ is an eigenvector of the matrix
  
  ${\displaystyle {\begin{pmatrix}-8&0&-1\\2&4&-5\\-3&7&2\end{pmatrix}}.}$
  
  In this case, determine the corresponding eigenvalue.
  
  ### Exercise (1 mark)
  
  Check whether the vector ${\displaystyle {}{\begin{pmatrix}7\\2\\3\end{pmatrix}}}$ is an eigenvector of the matrix
  
  ${\displaystyle {\begin{pmatrix}2&-4&-9\\-2&7&-2\\-1&-1&0\end{pmatrix}}.}$
  
  In this case, determine the corresponding eigenvalue.
  
  ### Exercise (4 (1+3) marks)
  
  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) ${\displaystyle {}1/10}$ of the people in bed go to the Nightowl, ${\displaystyle {}1/12}$ go to the Pirouette and the rest stays in bed.
  
  b) ${\displaystyle {}1/3}$ of the people in the Nightowl go to the Pirouette, ${\displaystyle {}1/5}$ go to bed and the rest stays in the Nightowl.
  
  c) ${\displaystyle {}3/5}$ of the people in the Pirouette stay in the Pirouette, a percentage of ${\displaystyle {}8}$ go to the Nightowl, the rest goes to bed.
  
  1. Establish a matrix, which describes the movements within an hour.
  2. Kleineisenstein has ${\displaystyle {}500}$ inhabitants. Determine a distribution of the inhabitants (on the three locations) which does not change within an hour.
  
  ### Exercise (3 marks)
  
  Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}\varphi \colon V\rightarrow V}$ be an endomorphism on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$. Show that ${\displaystyle {}\varphi }$ is a homothety if and only if every vector ${\displaystyle {}v\in V}$, ${\displaystyle {}v\neq 0}$, is an eigenvector of ${\displaystyle {}\varphi }$.
  
  ### Exercise (4 marks)
  
  Consider the matrix
  
  ${\displaystyle {}M={\begin{pmatrix}1&1\\-1&1\end{pmatrix}}\,.}$
  
  Show that ${\displaystyle {}M}$, as a real matrix, has no eigenvalue. Determine the eigenvalues and the eigenspaces of ${\displaystyle {}M}$ as a complex matrix.
  
  ### Exercise (6 marks)
  
  Consider the real matrices
  
  ${\displaystyle {}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \operatorname {Mat} _{2}(\mathbb {R} )\,.}$
  
  Characterize, in dependence on ${\displaystyle {}a,b,c,d}$, when such a matrix has
  
  1. two different eigenvalues,
  2. one eigenvalue with a two-dimensional eigenspace,
  3. one eigenvalue with a one-dimensional eigenspace,
  4. no eigenvalue.
  
  Footnotes
  1. The value ${\displaystyle {}n=0}$ is allowed, but does not say much.
  2. The expression ${\displaystyle P(\varphi )}$ means that the linear mapping ${\displaystyle {}\varphi }$ is inserted into the polynomial ${\displaystyle {}P}$. Here, ${\displaystyle {}X^{n}}$ has to be interpreted as ${\displaystyle {}\varphi ^{n}}$, the ${\displaystyle {}n}$-th composition of ${\displaystyle {}\varphi }$ with itself. The addition becomes the addition of linear mappings, etc.
  3. In this context, one also says eigenfunction instead of eigenvector.
explanation: The document is a collection of exercises from a mathematics course on applied sciences, specifically focusing on linear algebra and eigenvalues/eigenvectors. Each exercise presents a problem or a statement to prove, and there are no provided answers. The exercises cover various topics such as determining eigenvectors, eigenvalues, and eigenspaces for different linear mappings and matrices, as well as properties of eigenvalues and eigenvectors.
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distilabel_metadata.content: # Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 27
  
  Exercises
  
  ### Exercise
  
  Determine the eigenvectors of the function ${\displaystyle {}\mathbb {R} \rightarrow \mathbb {R} ,x\mapsto \pi x}$.
  
  ### Exercise
  
  Check whether the vector ${\displaystyle {}{\begin{pmatrix}3\\1\\-1\end{pmatrix}}}$ is an eigenvector for the matrix
  
  ${\displaystyle {\begin{pmatrix}-2&-5&1\\0&-2&2\\4&-3&5\end{pmatrix}}.}$
  
  In this case, determine the corresponding eigenvalue.
  
  ### Exercise
  
  Determine the eigenvectors and the eigenvalues for a linear mapping
  
  ${\displaystyle \varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2},}$
  
  given by a matrix of the form ${\displaystyle {}{\begin{pmatrix}a&b\\0&d\end{pmatrix}}}$.
  
  ### Exercise
  
  Show that the first standard vector is an eigenvector for every upper triangular matrix. What is its eigenvalue?
  
  ### Exercise
  
  Let
  
  ${\displaystyle {}M={\begin{pmatrix}d_{1}&\ast &\cdots &\cdots &\ast \\0&d_{2}&\ast &\cdots &\ast \\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1}&\ast \\0&\cdots &\cdots &0&d_{n}\end{pmatrix}}\,}$
  
  be an upper triangular matrix. Show that an eigenvalue of ${\displaystyle {}M}$ is a diagonal entry of ${\displaystyle {}M}$.
  
  ### Exercise
  
  Determine the eigenvalues, eigenvectors and eigenspaces for a plane rotation ${\displaystyle {}{\begin{pmatrix}\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}}$, with rotation angle ${\displaystyle {}\alpha }$, ${\displaystyle {}0\leq \alpha <2\pi }$, over ${\displaystyle {}\mathbb {R} }$.
  
  ### Exercise
  
  Show that every matrix ${\displaystyle {}M\in \operatorname {Mat} _{2}(\mathbb {C} )}$ has at least one eigenvalue.
  
  ### Exercise
  
  Let
  
  ${\displaystyle \varphi ,\psi \colon V\longrightarrow V}$
  
  be an endomorphisms on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$, and let ${\displaystyle {}v\in V}$ be an eigenvector of ${\displaystyle {}\varphi }$ and of ${\displaystyle {}\psi }$. Show that ${\displaystyle {}v}$ is also an eigenvector of ${\displaystyle {}\varphi \circ \psi }$. What is its eigenvalue?
  
  ### Exercise
  
  Let ${\displaystyle {}\varphi \colon V\rightarrow V}$ be an isomorphism on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$, and let ${\displaystyle {}\varphi ^{-1}}$ be its inverse mapping. Show that ${\displaystyle {}a\in K}$ is an eigenvalue of ${\displaystyle {}\varphi }$ if and only if ${\displaystyle {}a^{-1}}$ is an eigenvalue of ${\displaystyle {}\varphi ^{-1}}$.
  
  ### Exercise
  
  Give an example of a linear mapping
  
  ${\displaystyle \varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}}$
  
  such that ${\displaystyle {}\varphi }$ has no eigenvalue, but a certain power ${\displaystyle {}\varphi ^{n}}$, ${\displaystyle {}n\geq 2}$, has an eigenvalue.
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}\varphi \colon V\rightarrow V}$ be an endomorphism on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$, satisfying
  
  ${\displaystyle {}\varphi ^{n}=\operatorname {Id} _{V}\,}$
  
  for a certain ${\displaystyle {}n\in \mathbb {N} }$.[1] Show that every eigenvalue ${\displaystyle {}\lambda }$ of ${\displaystyle {}\varphi }$ fulfills the property ${\displaystyle {}\lambda ^{n}=1}$.
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}\varphi \colon V\rightarrow V}$ be an endomorphism on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$. Let ${\displaystyle {}\lambda \in K}$ be an eigenvalue of ${\displaystyle {}\varphi }$ and ${\displaystyle {}P\in K[X]}$ a polynomial. Show that ${\displaystyle {}P(\lambda )}$ is an eigenvalue[2] of ${\displaystyle {}P(\varphi )}$.
  
  ### Exercise
  
  Let ${\displaystyle {}M}$ be a square matrix, which can be written as a block matrix
  
  ${\displaystyle {}M={\begin{pmatrix}A&0\\0&B\end{pmatrix}}\,}$
  
  with square matrices ${\displaystyle {}A}$ and ${\displaystyle {}B}$. Show that a number ${\displaystyle {}\lambda \in K}$ is an eigenvalue of ${\displaystyle {}M}$ if and only if ${\displaystyle {}\lambda }$ is an eigenvalue of ${\displaystyle {}A}$ or of ${\displaystyle {}B}$.
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, ${\displaystyle {}V}$ a ${\displaystyle {}K}$-vector space and
  
  ${\displaystyle \varphi \colon V\longrightarrow V}$
  
  a linear mapping. Show that the following statements hold.
  
  1. Every eigenspace
  ${\displaystyle \operatorname {Eig} _{\lambda }{\left(\varphi \right)}}$
  
  is a linear subspace of ${\displaystyle {}V}$.
  
  2. ${\displaystyle {}\lambda }$ is an eigenvalue for ${\displaystyle {}\varphi }$ if and only if the eigenspace ${\displaystyle {}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}}$ is not the nullspace.
  3. A vector ${\displaystyle {}v\in V,\,v\neq 0}$, is an eigenvector for ${\displaystyle {}\lambda }$ if and only if ${\displaystyle {}v\in \operatorname {Eig} _{\lambda }{\left(\varphi \right)}}$.
  
  ### Exercise
  
  Let ${\displaystyle {}V=\mathbb {R} [X]_{\leq d}}$ denote the set of all real polynomials of degree ${\displaystyle {}\leq d}$. Determine the eigenvalues, eigenvectors and eigenspaces of the derivation operator
  
  ${\displaystyle V\longrightarrow V,P\longmapsto P'.}$
  
  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
  
  Let ${\displaystyle {}V}$ denote the real vector space, which consists of all functions from ${\displaystyle {}\mathbb {R} }$ tp ${\displaystyle {}\mathbb {R} }$, which are arbitrarily often differentiable.
  
  a) Show that the derivation ${\displaystyle {}f\mapsto f'}$ is a linear mapping from ${\displaystyle {}V}$ to ${\displaystyle {}V}$.
  
  b) Determine the eigenvalues of the derivation and determine, for each eigenvalue, at least one eigenvector.[3]
  
  c) Determine for every real number the eigenspace and its dimension.
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, ${\displaystyle {}V}$ a ${\displaystyle {}K}$-vector space and
  
  ${\displaystyle \varphi \colon V\longrightarrow V}$
  
  a linear mapping. Show that
  
  ${\displaystyle {}\operatorname {ker} {\left(\varphi \right)}=\operatorname {Eig} _{0}{\left(\varphi \right)}\,.}$
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}\varphi \colon V\rightarrow V}$ be an endomorphism on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$. Let ${\displaystyle {}\lambda \in K}$ and let
  
  ${\displaystyle {}U=\operatorname {Eig} _{\lambda }{\left(\varphi \right)}\,}$
  
  be the corresponding eigenspace. Show that ${\displaystyle {}\varphi }$ can be restricted to a linear mapping
  
  ${\displaystyle \varphi {|}_{U}\colon U\longrightarrow U,v\longmapsto \varphi (v),}$
  
  and that this mapping is the homothety with scale factor ${\displaystyle {}\lambda }$.
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, ${\displaystyle {}V}$ a ${\displaystyle {}K}$-vector space and
  
  ${\displaystyle \varphi \colon V\longrightarrow V}$
  
  a linear mapping. Let ${\displaystyle {}\lambda _{1}\neq \lambda _{2}}$ be elements in ${\displaystyle {}K}$. Show that
  
  ${\displaystyle {}\operatorname {Eig} _{\lambda _{1}}{\left(\varphi \right)}\cap \operatorname {Eig} _{\lambda _{2}}{\left(\varphi \right)}=0\,.}$
  
  ### Exercise
  
  Let ${\displaystyle {}K}$ be a field, ${\displaystyle {}V}$ a finite-dimensional ${\displaystyle {}K}$-vector space and
  
  ${\displaystyle \varphi \colon V\longrightarrow V}$
  
  a linear mapping. Show that there exist at most ${\displaystyle {}\dim _{}{\left(V\right)}}$ many eigenvalues for ${\displaystyle {}\varphi }$.
  
  Hand-in-exercises
  
  ### Exercise (1 mark)
  
  Check whether the vector ${\displaystyle {}{\begin{pmatrix}6\\-5\\8\end{pmatrix}}}$ is an eigenvector of the matrix
  
  ${\displaystyle {\begin{pmatrix}-8&0&-1\\2&4&-5\\-3&7&2\end{pmatrix}}.}$
  
  In this case, determine the corresponding eigenvalue.
  
  ### Exercise (1 mark)
  
  Check whether the vector ${\displaystyle {}{\begin{pmatrix}7\\2\\3\end{pmatrix}}}$ is an eigenvector of the matrix
  
  ${\displaystyle {\begin{pmatrix}2&-4&-9\\-2&7&-2\\-1&-1&0\end{pmatrix}}.}$
  
  In this case, determine the corresponding eigenvalue.
  
  ### Exercise (4 (1+3) marks)
  
  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) ${\displaystyle {}1/10}$ of the people in bed go to the Nightowl, ${\displaystyle {}1/12}$ go to the Pirouette and the rest stays in bed.
  
  b) ${\displaystyle {}1/3}$ of the people in the Nightowl go to the Pirouette, ${\displaystyle {}1/5}$ go to bed and the rest stays in the Nightowl.
  
  c) ${\displaystyle {}3/5}$ of the people in the Pirouette stay in the Pirouette, a percentage of ${\displaystyle {}8}$ go to the Nightowl, the rest goes to bed.
  
  1. Establish a matrix, which describes the movements within an hour.
  2. Kleineisenstein has ${\displaystyle {}500}$ inhabitants. Determine a distribution of the inhabitants (on the three locations) which does not change within an hour.
  
  ### Exercise (3 marks)
  
  Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}\varphi \colon V\rightarrow V}$ be an endomorphism on a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$. Show that ${\displaystyle {}\varphi }$ is a homothety if and only if every vector ${\displaystyle {}v\in V}$, ${\displaystyle {}v\neq 0}$, is an eigenvector of ${\displaystyle {}\varphi }$.
  
  ### Exercise (4 marks)
  
  Consider the matrix
  
  ${\displaystyle {}M={\begin{pmatrix}1&1\\-1&1\end{pmatrix}}\,.}$
  
  Show that ${\displaystyle {}M}$, as a real matrix, has no eigenvalue. Determine the eigenvalues and the eigenspaces of ${\displaystyle {}M}$ as a complex matrix.
  
  ### Exercise (6 marks)
  
  Consider the real matrices
  
  ${\displaystyle {}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \operatorname {Mat} _{2}(\mathbb {R} )\,.}$
  
  Characterize, in dependence on ${\displaystyle {}a,b,c,d}$, when such a matrix has
  
  1. two different eigenvalues,
  2. one eigenvalue with a two-dimensional eigenspace,
  3. one eigenvalue with a one-dimensional eigenspace,
  4. no eigenvalue.
  
  Footnotes
  1. The value ${\displaystyle {}n=0}$ is allowed, but does not say much.
  2. The expression ${\displaystyle P(\varphi )}$ means that the linear mapping ${\displaystyle {}\varphi }$ is inserted into the polynomial ${\displaystyle {}P}$. Here, ${\displaystyle {}X^{n}}$ has to be interpreted as ${\displaystyle {}\varphi ^{n}}$, the ${\displaystyle {}n}$-th composition of ${\displaystyle {}\varphi }$ with itself. The addition becomes the addition of linear mappings, etc.
  3. In this context, one also says eigenfunction instead of eigenvector.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a collection of exercises from a mathematics course on applied sciences, specifically focusing on linear algebra and eigenvalues/eigenvectors. Each exercise presents a problem or a statement to prove, and there are no provided answers. The exercises cover various topics such as determining eigenvectors, eigenvalues, and eigenspaces for different linear mappings and matrices, as well as properties of eigenvalues and eigenvectors.
  
  Label: multiple-question