Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 53

Exercises

Exercise

Let ${}V$ and ${}W$ be finite-dimensional normed ${}{\mathbb {K} }$-vector spaces. Show that the maximum norm on the homomorphism space ${}\operatorname {Hom} _{K}{\left(V,W\right)}$ is indeed a norm.

Exercise

Let ${}V$ and ${}W$ denote Euclidean vector spaces, and let

$\varphi \colon V\longrightarrow W$

be a linear mapping. Show that there exists a vector ${}v\in V$, ${}\Vert {v}\Vert =1$, such that

${}\Vert {\varphi (v)}\Vert =\Vert {\varphi }\Vert \,$

holds.

Exercise

Compute for the matrix

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

a) the maximum norm, the sum norm, and the Euclidean norm,

b) the maximum norm for the maximum norm, the sum norm, or the Euclidean norm on ${}\mathbb {R} ^{2}$ in all combinations,

c) the column sum norm and the row sum norm.

Exercise

Show that the column sum norm on the space of matrices ${}\operatorname {Mat} _{m\times n}({\mathbb {K} })$ equals the maximum norm in the sense of Definition, when we endow the spaces ${}{\mathbb {K} }^{n}$ and ${}{\mathbb {K} }^{m}$ with the sum norm.

Exercise

We consider the linear mapping

$\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2},(x,y)\longmapsto {\begin{pmatrix}1&0\\1&0\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}},$

where ${}\mathbb {R} ^{2}$ is endowed with the Euclidean norm. Determine the eigenvalues, the eigenvectors, and the norm of ${}\varphi$.

Exercise

Let

$\varphi \colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ,(x_{1},\ldots ,x_{n})\longmapsto \sum _{i=1}^{n}a_{i}x_{i},$

be a linear mapping ${}\neq 0$. Determine a vector ${}v\in \mathbb {R} ^{n}$ on the closed ball with center ${}0$ and radius ${}1$ where the function

$B\left(0,1\right)\longrightarrow \mathbb {R} ,v\longmapsto \vert {\varphi (v)}\vert ,$

obtain its maximum. Determine the norm of ${}\varphi$.

Exercise

Show that the matrix multiplication

$\operatorname {Mat} _{m\times n}({\mathbb {K} })\times \operatorname {Mat} _{n\times p}({\mathbb {K} })\longrightarrow \operatorname {Mat} _{m\times p}({\mathbb {K} }),(A,B)\longmapsto A\circ B,$

is continuous.

Let

$f\colon L\longrightarrow M,x\longmapsto f(x),$

denote a mapping between the metric spaces ${}L$ and ${}M$ The mapping is called Lipschitz continuous if there exists a real number ${}c\geq 0$ such that

${}d{\left(f(x),f(y)\right)}\leq c\cdot d{\left(x,y\right)}\,$

holds for all

${}x,y\in L$.

Exercise

Let ${}V$ and ${}W$ be finite-dimensional normed ${}{\mathbb {K} }$-vector spaces, and let

$\varphi \colon V\longrightarrow W$

denote a linear mapping. Show that ${}\varphi$ is Lipschitz continuous.

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, and let ${}\varphi \colon V\rightarrow V$ denote a linear mapping.

a) Show that the estimate

${}\Vert {\varphi (v)}\Vert \leq \Vert {v}\Vert \,$

holds for every vector ${}v\in V$ if and only if the supremum norm fulfills

${}\Vert {\varphi }\Vert \leq 1\,.$

b) Show that ${}\varphi$ is stable if it satisfies the condition from part (a).

c) Give an example for a ${}\varphi$ that is stable but does not fulfill the condition from part (a).

Exercise

Show that a linear mapping

$\varphi \colon V\longrightarrow W$

between finite-dimensional normed ${}{\mathbb {K} }$-vector spaces ${}V$ and ${}W$ is a contraction if and only if ${}\Vert {\varphi }\Vert <1$ holds.

Exercise

Let

${}M={\begin{pmatrix}{\mathrm {i} }&1\\0&{\mathrm {i} }\end{pmatrix}}\,.$

a) Establish a formula for ${}M^{n}$.

b) Is the sequence ${}M^{n}$ bounded? Is it convergent?

Exercise

Establish a formula for the powers

${\begin{pmatrix}{\frac {1}{2}}&1&0\\0&{\frac {1}{2}}&1\\0&0&{\frac {1}{2}}\end{pmatrix}}^{n}.$

Exercise

Let

${}W=\operatorname {End} _{}{\left(V\right)}\,$

be the endomorphism space for a finite-dimensional ${}{\mathbb {K} }$-vector space ${}V$. Which properties of a norm does the spectral radius ${}\varphi \mapsto \rho (\varphi )$ have, which not?

Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, and let ${}v_{n}$, ${}n\in \mathbb {N}$, denote a sequence in ${}V$. Show that the sequence converges (with respect to an arbitrary norm) if and only if for some (every) basis all component sequences converge in ${}{\mathbb {K} }$.

Exercise

Show that a nilpotent endomorphism

$\varphi \colon V\longrightarrow V$

on a ${}{\mathbb {K} }$-vector space ${}V$ is asymptotically stable.

Exercise

Show that an endomorphism

$\varphi \colon V\longrightarrow V$

of finite order on a finite-dimensional ${}{\mathbb {K} }$-vector space ${}V$ is stable.

Exercise

Show, aling every characterization of Theorem 53.10 , that an isometry ${}\varphi \colon V\rightarrow V$ on a Euclidean vector space ${}V$ is stable.

Exercise

Let

${}V=U\oplus W\,$

be a direct sum decomposition of a finite-dimensional ${}{\mathbb {K} }$-vector space ${}V$. Let

$\varphi \colon V\longrightarrow V$

be an endomorphism with the direct sum decomposition

${}\varphi =\psi \oplus \theta \,.$

Show that ${}\varphi$ is asymptotically stable if and only if ${}\psi$ ane ${}\theta$ are asymptotically stable.

Exercise

Let

${}V=U\oplus W\,$

be a direct sum decomposition of a finite-dimensional ${}{\mathbb {K} }$-vector space ${}V$. Let

$\varphi \colon V\longrightarrow V$

be an endomorphism with the direct sum decomposition

${}\varphi =\psi \oplus \theta \,.$

Show that ${}\varphi$ is stable if and only if ${}\psi$ and ${}\theta$ are stable.

Exercise

Let

${}V=U\oplus W\,$

be a direct sum decomposition of a finite-dimensional ${}{\mathbb {K} }$-vector space ${}V$. Let

$\varphi \colon V\longrightarrow V$

be an endomorphism with the direct sum decomposition

${}\varphi =\psi \oplus \theta \,.$

Show that the sequence ${}\varphi ^{n}$ converges if and only if the sequences ${}\psi ^{n}$ and ${}\theta ^{n}$ converge.

Exercise

Show the equality

${}{\begin{pmatrix}\lambda &1&0&\cdots &0\\0&\lambda &1&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\\vdots &\cdots &0&\lambda &1\\0&\cdots &\cdots &0&\lambda \end{pmatrix}}^{n}{\begin{pmatrix}1\\1\\0\\\vdots \\0\end{pmatrix}}={\begin{pmatrix}\lambda ^{n}+n\lambda ^{n-1}\\\lambda ^{n}\\0\\\vdots \\0\end{pmatrix}}\,.$

Exercise

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

$\varphi \colon V\longrightarrow V$

be an endomorphism. Show that the following properties are equivalent.

1. The sequence ${}\varphi ^{n}$ converges in ${}\operatorname {End} _{}{\left(V\right)}$.
2. For every ${}v\in V$, the sequence ${}\varphi ^{n}(v)$, ${}n\in \mathbb {N}$ converges
3. There exists a generating system ${}v_{1},\ldots ,v_{m}\in V$ such that ${}\varphi ^{n}(v_{j})$, ${}j=1,\ldots ,m$, converges.
4. The modulus of every complex eigenvalue of ${}\varphi$ issmaller or equal ${}1$, and if its modulus is ${}1$, then the eigenvalue equals ${}1$, and it is diagonalizable.
5. For a describing matrix ${}M$ of ${}\varphi$, considered over ${}\mathbb {C}$, the Jordan blocks of the Jordan normal form are

   ${\begin{pmatrix}\lambda &1&0&\cdots &0\\0&\lambda &1&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\\vdots &\cdots &0&\lambda &1\\0&\cdots &\cdots &0&\lambda \end{pmatrix}}$

   with ${}\vert {\lambda }\vert <1$, or equal ${}(1)$.

Exercise

Let ${}V$ and ${}W$ be finite-dimensional normed ${}{\mathbb {K} }$-vector spaces, and let

$\psi \colon V\longrightarrow W$

denote an isomorphism. Let

$f\colon V\longrightarrow V$

be an endomorphism, and let

${}g:=\psi \circ f\circ \psi ^{-1}\,$

be the corresponding endomorphism on ${}W$. Show that ${}f$ is stable (asymptotically stable) if and only if ${}g$ satisfies this condition.

Exercise

Let ${}\varphi \colon V\rightarrow V$ be an endomorphism of a finite-dimensional ${}{\mathbb {K} }$-vector space ${}V$. Show the following statements.

a) If ${}\varphi$ is asymptotically stable, then the sequence ${}{\left(\det \varphi \right)}^{n}$, ${}n\in \mathbb {N}$, converges zo ${}0$.

b) If ${}\varphi$ is stable, then the sequence ${}{\left(\det \varphi \right)}^{n}$, ${}n\in \mathbb {N}$, is bounded.

c) If the sequence ${}\varphi ^{n}$, ${}n\in \mathbb {N}$, converges, then the sequence ${}{\left(\det \varphi \right)}^{n}$, ${}n\in \mathbb {N}$, converges to ${}0$ or to ${}1$.

Exercise

Give an example of a matrix that is not stable, but such that the sequence ${}{\left(\det \varphi \right)}^{n}$, ${}n\in \mathbb {N}$, converges to ${}0$.

Let ${}T$ denote a set, ${}M$ a metric space, and

$f_{n}\colon T\longrightarrow M$

(${}n\in \mathbb {N}$) a sequence of mappings. We say that the sequence of mappings converges pointwise if for every ${}x\in T$ the sequence

${\left(f_{n}(x)\right)}_{n\in \mathbb {N} }$

converges

in ${}M$.

Exercise

Let ${}M_{k}={\left({\left(a_{ij}\right)}_{k}\right)}_{1\leq i\leq m,1\leq j\leq n}$ be a sequence of real ${}m\times n$-matrices, and let

$\varphi _{k}\colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m}$

denote the corresponding sequence of linear mappings. Show that the sequences of the entries ${}{\left({a_{ij}}\right)}_{k}$ converge for all ${}i,j$ if and only if the sequence of the mappings converge pointwise.

Hand-in-exercises

Exercise (2 marks)

Let ${}V$ and ${}W$ be finite-dimensional normed ${}{\mathbb {K} }$-vector spaces, and let ${}\varphi \colon V\rightarrow W$ denote a linear mapping. Show the estimate

${}\Vert {\varphi (v)}\Vert \leq \Vert {\varphi }\Vert \cdot \Vert {v}\Vert \,$

for all ${}v\in V$.

Exercise (11 (1+9+1) marks)

Compute for the matrix

${\begin{pmatrix}3&2\\0&1\end{pmatrix}}$

a) the maximum norm, the sum norm, and the Euclidean norm,

b) the maximum norm for the maximum norm, the sum norm, or the Euclidean norm on ${}\mathbb {R} ^{2}$ in all combinations,

c) the column sum norm and the row sum norm.

Exercise (3 marks)

Let ${}V$ denote a Euclidean vector space, and let

$\varphi \colon V\longrightarrow V$

be a linear mapping such that there exists an orthogonal basis of ${}V$ consisting of eigenvectors of ${}\varphi$. Show that

${}\Vert {\varphi }\Vert ={\max {\left(\vert {\lambda }\vert ,\lambda {\text{ is eigenvalue of }}\varphi \right)}}\,$

holds.

Exercise (6 marks)

Let ${}M$ be an ${}n\times n$-matrix over ${}{\mathbb {K} }$. Show that the following statements are equivalent.

a) In the sequence ${}M^{n}$, ${}n\in \mathbb {N}$, there is a repetion, that is,

${}M^{n}=M^{m}\,$

for some pair of numbers ${}n<m$.

b) In the sequence ${}M^{n}$, ${}n\in \mathbb {N}$, there only finitely many different matrices.

c) The sequence ${}M^{n}$, ${}n\in \mathbb {N}$, becomes eventually (starting at a certain position) periodic.

d) The Jordan blocks of ${}M$ over ${}\mathbb {C}$ have the form

${\begin{pmatrix}0&1&0&\cdots &0\\0&0&1&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\\vdots &\cdots &0&0&1\\0&\cdots &\cdots &0&0\end{pmatrix}},$

or the form ${}(\lambda )$ with a

complex root of unity ${}\lambda$.

Exercise (6 marks)

Suppose that the real plane ${}\mathbb {R} ^{2}$ is endowed either with the Euclidean metric, the sum metric, or the maximum metric. Determine, in dependence of the metric chosen, the maximal number of points ${}P_{1},\ldots ,P_{n}\in \mathbb {R} ^{2}$ such that the metric on the subset ${}T=\{P_{1},\ldots ,P_{n}\}$ induces the discrete metric.

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