Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 28

Exercises

Exercise

Compute the characteristic polynomial of the matrix

${\begin{pmatrix}2&5&3\\7&4&2\\3&7&5\end{pmatrix}}.$

Exercise

Compute the characteristic polynomial, the eigenvalues and the eigenspaces of the matrix

${\begin{pmatrix}5&7\\3&4\end{pmatrix}}$

over ${}\mathbb {C}$.

Exercise

Show that the characteristic polynomial of a linear mapping ${}\varphi \colon V\rightarrow V$ on a finite-dimensional ${}K$-vector space ${}V$ is well-defined, that is, independent of the chosen basis.

Exercise

Let ${}K$ be a field and let ${}M$ denote an ${}n\times n$-matrix over ${}K$. Show that for every ${}\lambda \in K$, the relation

${}\chi _{M}(\lambda )=\det {\left(\lambda E_{n}-M\right)}\,$

holds.

Exercise

Let ${}K$ be a field and let ${}M$ be an ${}n\times n$-matrix over ${}K$. Where can you find the determinant of ${}M$ within the characteristic polynomial ${}\chi _{M}$?

Exercise

Show that the characteristic polynomial of the so-called companion matrix

${}M={\begin{pmatrix}0&1&0&\ldots &0\\0&0&1&\ldots &0\\\vdots &\vdots &\ddots &\ddots &\vdots \\0&0&\ldots &0&1\\-a_{0}&-a_{1}&\ldots &-a_{n-2}&-a_{n-1}\end{pmatrix}}\,$

equals

${}\chi _{M}=X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}\,.$

Exercise

We consider the real matrix

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

a) Determine

$M^{n}{\begin{pmatrix}1\\0\end{pmatrix}}$

for ${}n=1,2,3,4$.

b) Let

${}{\begin{pmatrix}x_{n+1}\\y_{n+1}\end{pmatrix}}:=M^{n}{\begin{pmatrix}1\\0\end{pmatrix}}\,.$

Establish a relation between the sequences ${}x_{n}$ and ${}y_{n}$, and determine a recursive formula for these sequences.

c) Determine the eigenvalues and the eigenvectors of ${}M$.

Exercise

Let

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

1. Determine the characteristic polynomial of ${}M$.
2. Determine a zero of the characteristic polynomial of ${}M$, and write the polynomial using the corresponding linear factor.
3. Show that the characteristic polynomial of ${}M$ has at least two real roots.

Exercise

Let ${}\lambda$ be a zero of the polynomial

$X^{3}+2X^{2}-2.$

Show that

${\begin{pmatrix}{\frac {1}{(1+\lambda )^{3}}}\\{\frac {1}{(1+\lambda )^{2}}}\\{\frac {1}{(1+\lambda )}}\\1\end{pmatrix}}$

is an eigenvector of the matrix

${\begin{pmatrix}-1&1&0&0\\0&-1&1&0\\0&0&-1&1\\-1&0&0&1\end{pmatrix}}$

for the eigenvalue ${}\lambda$.

To solve the following exercise, the two exercises above and also Exercise 24.31 are helpful.

Exercise

We consider the mapping

$\Psi \colon \mathbb {R} _{\geq 0}^{4}\longrightarrow \mathbb {R} _{\geq 0}^{4},$

which assigns to a four tuple ${}(a,b,c,d)$ the four tuple

$(\vert {b-a}\vert ,\vert {c-b}\vert ,\vert {d-c}\vert ,\vert {a-d}\vert ).$

Show that there exists a tuple ${}(a,b,c,d)$, for which arbitrary iterations of the mapping do never reach the zero tuple.

Exercise

Determine the eigenvalues and the eigenspaces of the linear mapping

$\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3},v\longmapsto Mv,$

given by the matrix

${}M={\begin{pmatrix}2&0&5\\0&-1&0\\8&0&5\end{pmatrix}}\,.$

Exercise

We consider the linear mapping

$\varphi \colon \mathbb {C} ^{3}\longrightarrow \mathbb {C} ^{3},$

which is given by the matrix

${}A={\begin{pmatrix}2&1&-2+{\mathrm {i} }\\0&{\mathrm {i} }&1+{\mathrm {i} }\\0&0&-1+2{\mathrm {i} }\end{pmatrix}}\,,$

with respect to the standard basis.

a) Determine the characteristic polynomial and the eigenvalues of ${}A$.

b) Compute, for every eigenvalue, an eigenvector.

c) Establish a matrix for ${}\varphi$ with respect to a basis of eigenvectors.

Exercise

Let

${}A={\begin{pmatrix}-4&6&6\\0&2&0\\-3&3&5\end{pmatrix}}\in \operatorname {Mat} _{3\times 3}(\mathbb {R} )\,.$

Compute:

1. the eigenvalues of ${}A$;
2. the corresponding eigenspaces;
3. the geometric and algebraic multiplicities of each eigenvalue;
4. a matrix ${}C\in \operatorname {Mat} _{3\times 3}(\mathbb {R} )$ such that ${}C^{-1}AC$ is a diagonal matrix.

Exercise

Determine the eigenspace and the geometric multiplicity for ${}-2$ of the matrix

${\begin{pmatrix}2&1&3\\5&0&7\\9&3&8\end{pmatrix}}.$

Exercise

Show that the matrix

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

is diagonalizable over ${}\mathbb {Q}$.

Exercise

Let ${}M\in \operatorname {Mat} _{n}(K)$ be a matrix with ${}n$ (pairwise) different eigenvalues. Show that the determinant of ${}M$ is the product of the eigenvalues.

Exercise

Let ${}K$ be a field, ${}a\in K$ and ${}m,n\in \mathbb {N} _{+}$ numbers with ${}1\leq m\leq n$. Give an example of an ${}n\times n$-matrix ${}M$, such that ${}a$ is an eigenvalue for ${}M$ with algebraic multiplicity ${}n$ and geometric multiplicity ${}m$.

Exercise

Determine, which of the following elementary-geometric mappings are linear, which are diagonalizable and which are trigonalizable.

1. The reflection in the plane, given by the line ${}4x-7y=0$ as axis.
2. The translation with the vector ${}\left(5,\,-3\right)$.
3. The rotation by ${}30$ degree counter-clockwise around the origin.
4. The reflection with ${}(1,0)$ as center.

Exercise

Determine, whether the real matrix

${\begin{pmatrix}4&7&-3\\2&7&5\\0&0&-6\end{pmatrix}}$

is trigonalizable or not.

Exercise

Suppose that a linear mapping

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

is given by the matrix

${\begin{pmatrix}3&5\\0&3\end{pmatrix}}$

with respect to the standard basis. Find a basis, such that ${}\varphi$ is described by the matrix

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

with respect to this basis.

The next exercises use the following definition.

Let ${}K$ be a field, ${}V$ a vector space over ${}K$ and

$\varphi \colon V\longrightarrow V$

a linear mapping. A linear subspace ${}U\subseteq V$ is called ${}\varphi$-invariant, if

${}\varphi (U)\subseteq U\,$

holds.

Exercise

Let ${}\varphi \colon V\rightarrow V$ a linear mapping on a ${}K$-vector space ${}V$ over a field ${}K$. Show the following properties.

1. The zero space ${}0\subseteq V$ is ${}\varphi$-invariant.
2. ${}V$ is ${}\varphi$-invariant.
3. Eigenspaces are ${}\varphi$-invariant.
4. Let ${}U_{1},U_{2}\subseteq V$ be ${}\varphi$-invariant linear subspaces. Then also ${}U_{1}\cap U_{2}$ and ${}U_{1}+U_{2}$ are ${}\varphi$-invariant.
5. Let ${}U\subseteq V$ be a ${}\varphi$-invariant linear subspace. Then also the image space ${}\varphi (U)$ and the preimage space ${}\varphi ^{-1}(U)$ are ${}\varphi$-invariant.

Exercise

Let ${}\varphi \colon V\rightarrow V$ a linear mapping on a ${}K$-vector space ${}V$ over a field ${}K$, and let ${}v\in V$. Show that the smallest ${}\varphi$-invariant linear subspace of ${}V$, which contains ${}v$, equals

$\langle \varphi ^{n}(v),\,n\in \mathbb {N} \rangle .$

Exercise

Let ${}\varphi \colon V\rightarrow V$ a linear mapping on a ${}K$-vector space ${}V$ over a field ${}K$. Show that the subset of ${}V$, defined by

${}U={\left\{v\in V\mid {\text{ there exists an }}n\in \mathbb {N} {\text{ with }}\varphi ^{n}(v)=0\right\}}\,,$

is an ${}\varphi$-invariant linear subspace.

Exercise

Let ${}\varphi \colon V\rightarrow V$ be a linear mapping on a ${}K$-vector space ${}V$. Let ${}v_{1},\ldots ,v_{n}$ be a basis of ${}V$, such that ${}\varphi$ is described, with respect to this basis, by an upper triangular matrix. Show that the linear subspaces

$\langle v_{1},\ldots ,v_{i}\rangle$

are ${}\varphi$-invariant for every ${}i$.

Exercise

Determine, whether the real matrix

${\begin{pmatrix}-4&-1&-2&3\\6&7&7&1\\0&0&3&-2\\0&0&6&2\end{pmatrix}}$

is trigonalizable or not.

Hand-in-exercises

Exercise (2 marks)

Compute the characteristic polynomial of the matrix

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

Exercise (3 marks)

Compute the characteristic polynomial, the eigenvalues and the eigenspaces of the matrix

${\begin{pmatrix}2&7\\5&4\end{pmatrix}}$

over ${}\mathbb {C}$.

Exercise (4 marks)

Let

$\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$

be a linear mapping. Show that ${}\varphi$ has at least one eigenvector.

Exercise (4 marks)

Let

${}A={\begin{pmatrix}-5&0&7\\6&2&-6\\-4&0&6\end{pmatrix}}\in \operatorname {Mat} _{3\times 3}(\mathbb {R} )\,.$

Compute:

1. the eigenvalues of ${}A$;
2. the corresponding eigenspaces;
3. the geometric and algebraic multiplicities of each eigenvalue;
4. a matrix ${}C\in \operatorname {Mat} _{3\times 3}(\mathbb {R} )$ such that ${}C^{-1}AC$ is a diagonal matrix.

Exercise (4 marks)

Determine for every ${}\lambda \in \mathbb {Q}$ the algebraic and geometric multiplicities for the matrix

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

Exercise (4 marks)

Decide whether the matrix

${\begin{pmatrix}5&-1&3\\7&9&8\\6&2&-7\end{pmatrix}}$

is trigonalizable over ${}\mathbb {R}$.

Exercise (3 marks)

Determine whether the real matrix

${\begin{pmatrix}1&5&6&2\\5&7&-4&-3\\0&0&-2&8\\0&0&1&9\end{pmatrix}}$

is trigonalizable or not.

Footnotes

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