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.