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

Exercises

### Exercise

Compute the characteristic polynomial of the matrix

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

### Exercise

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

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

over ${\displaystyle {}\mathbb {C} }$.

### Exercise

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

### Exercise

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

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

holds.[1]

### Exercise

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

### Exercise

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

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

equals

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

### Exercise

We consider the real matrix

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

a) Determine

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

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

b) Let

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

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

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

### Exercise

Let

${\displaystyle {}M={\begin{pmatrix}-1&1&0&0\\0&-1&1&0\\0&0&-1&1\\-1&0&0&1\end{pmatrix}}\,.}$
1. Determine the characteristic polynomial of ${\displaystyle {}M}$.
2. Determine a zero of the characteristic polynomial of ${\displaystyle {}M}$, and write the polynomial using the corresponding linear factor.
3. Show that the characteristic polynomial of ${\displaystyle {}M}$ has at least two real roots.

### Exercise

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

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

Show that

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

is an eigenvector of the matrix

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

for the eigenvalue ${\displaystyle {}\lambda }$.

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

### Exercise

We consider the mapping

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

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

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

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

### Exercise

Determine the eigenvalues and the eigenspaces of the linear mapping

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

given by the matrix

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

### Exercise

We consider the linear mapping

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

which is given by the matrix

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

with respect to the standard basis.

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

b) Compute, for every eigenvalue, an eigenvector.

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

### Exercise

Let

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

Compute:

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

### Exercise

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

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

### Exercise

Show that the matrix

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

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

### Exercise

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

### Exercise

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

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

### Exercise

Determine, whether the real matrix

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

is trigonalizable or not.

### Exercise

Suppose that a linear mapping

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

is given by the matrix

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

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

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

with respect to this basis.

The next exercises use the following definition.

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

${\displaystyle \varphi \colon V\longrightarrow V}$

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

${\displaystyle {}\varphi (U)\subseteq U\,}$
holds.

### Exercise

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

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

### Exercise

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

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

### Exercise

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

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

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

### Exercise

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

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

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

### Exercise

Determine, whether the real matrix

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

is trigonalizable or not.

Hand-in-exercises

### Exercise (2 marks)

Compute the characteristic polynomial of the matrix

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

### Exercise (3 marks)

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

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

over ${\displaystyle {}\mathbb {C} }$.

### Exercise (4 marks)

Let

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

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

### Exercise (4 marks)

Let

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

Compute:

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

### Exercise (4 marks)

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

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

### Exercise (4 marks)

Decide whether the matrix

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

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

### Exercise (3 marks)

Determine whether the real matrix

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

is trigonalizable or not.

Footnotes
1. The main difficulty might be here to recognize that there is indeed something to show.