Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 21/refcontrol

Exercise for the break

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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

Exercises

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Let a linear mappingMDLD/linear mapping

$\varphi \colon K\longrightarrow K$

be given. What are the eigenvaluesMDLD/eigenvalues and the eigenvectorsMDLD/eigenvectors of ${}\varphi$?

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

===Exercise Exercise 21.9

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

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

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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 ${}\lambda \in K$ 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 ${}\lambda$.

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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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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 ${}V_{1},\ldots ,V_{n}$ denote vector spacesMDLD/vector spaces over a fieldMDLD/field ${}K$, and let

$\varphi _{i}\colon V_{i}\longrightarrow V_{i}$

denote linear mappings.MDLD/linear mappings Let ${}a\in K$ be an eigenvalueMDLD/eigenvalue of ${}\varphi _{k}$ for a determined ${}k$. Show that ${}a$ is also an eigenvalue of the product mappingMDLD/product mapping

$\varphi _{1}\times \cdots \times \varphi _{n}\colon V_{1}\times \cdots \times V_{n}\longrightarrow V_{1}\times \cdots \times V_{n}.$

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Show that ${}\lambda \in K$ is an eigenvalueMDLD/eigenvalue for the linear mappingMDLD/linear mapping

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

given by a matrix of the form ${}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ if and only if ${}\lambda$ is a zeroMDLD/zero (function) of the polynomial

$X^{2}-(a+d)X+ad-cb.$

The concept of an eigenvector is also defined for infinite-dimensional vector spaces, and it is also important in this context, as the following exercise shows.

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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)

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Let

$\varphi \colon V\longrightarrow V$

be an endomorphismMDLD/endomorphism on a finite-dimensionalMDLD/finite-dimensional ${}K$-vector spaceMDLD/vector space ${}V$, and let ${}v\in V$ be an eigenvectorMDLD/eigenvector for ${}\varphi$ with eigenvalueMDLD/eigenvalue ${}\lambda \in K$. Let

${\varphi }^{*}\colon {V}^{*}\longrightarrow {V}^{*}$

be the dual mappingMDLD/dual mapping of ${}\varphi$. We consider bases of ${}V$ of the form ${}v,u_{1},\ldots ,u_{r}$ with the dual basis ${}v^{*},u_{1}^{*},\ldots ,u_{r}^{*}$. Give examples of the following behavior.

a) ${}v^{*}$ is an eigenvector of ${}{\varphi }^{*}$ with the eigenvalue ${}\lambda$ independent of ${}u_{1},\ldots ,u_{r}$.

b) ${}v^{*}$ is an eigenvector of ${}{\varphi }^{*}$ with the eigenvalue ${}\lambda$ with respect to some basis ${}v,u_{1},\ldots ,u_{r}$, but not with respect to another basis ${}v,w_{1},\ldots ,w_{r}$.

c) ${}v^{*}$ is for no basis ${}v,u_{1},\ldots ,u_{r}$ an eigenvector of ${}{\varphi }^{*}$.

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Let ${}V$ be a finite-dimensionalMDLD/finite-dimensional ${}K$-vector space,MDLD/vector space and ${}v\in V$, ${}v\neq 0$, a fixed vector. Show that

${}R(v)={\left\{\varphi \in \operatorname {End} _{}{\left(V\right)}\mid v{\text{ is eigenvector of }}\varphi \right\}}\,,$

with the natural addition and multiplication of endomorphisms, is a ringMDLD/ring and a linear subspaceMDLD/linear subspace of ${}\operatorname {End} _{}{\left(V\right)}$. Determine the dimensionMDLD/dimension (vs) of this space.

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Let ${}K$ be a field,MDLD/field ${}c\in K$ and let ${}a={\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\in K^{n}$ denote a vector different from ${}0$. Establish an inhomogeneous system of linear equationsMDLD/inhomogeneous system of linear equations such that its solution setMDLD/solution set (linear system) is the set of all ${}n\times n$-matrices,MDLD/matrices for that ${}a$ is an eigenvectorMDLD/eigenvector with eigenvalueMDLD/eigenvalue ${}c$. What is special about this system, and what is the dimension of its solution set?

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Let ${}M$ be a realMDLD/real ${}n\times n$-matrix.MDLD/matrix Let ${}a\in \mathbb {R}$ be a real number, and suppose that it is an eigenvalueMDLD/eigenvalue of ${}M$, considered as a complex matrix. Show that ${}a$ is already over ${}\mathbb {R}$ an eigenvalue of ${}M$.

Generalize the previous statement for any field extensionMDLD/field extension ${}K\subseteq L$.

Hand-in-exercises

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.

Exercise (2 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 ${}V$, satisfying

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

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

Exercise (4 marks) Create referencenumber

Let ${}K$ be a field,MDLD/field and let ${}V$ denote an ${}n$-dimensionalMDLD/dimensional (fgvs) ${}K$-vector space.MDLD/vector space Let

$\varphi \colon V\longrightarrow V$

be a linear mapping.MDLD/linear mapping Let ${}\lambda \neq 0$ be an eigenvalueMDLD/eigenvalue of ${}\varphi$, and ${}v$ a corresponding eigenvector.MDLD/eigenvector Show that, for a given basisMDLD/basis (vs) ${}v,u_{2},\ldots ,u_{n}$ of ${}V$, there exists a basis ${}v,w_{2},\ldots ,w_{n}$ such that ${}\langle v,u_{j}\rangle =\langle v,w_{j}\rangle$ and such that

${}\varphi (w_{j})\in \langle u_{i},\,i=2,\ldots ,n\rangle \,$

for all ${}j=2,\ldots ,n$ holds.

Show also that this is not possible for ${}\lambda =0$.

Footnotes

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