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

Exercises

Exercise

Compute the characteristic polynomial of the matrix

$\left(\begin{array}{ccc}2& 5& 3\\ 7& 4& 2\\ 3& 7& 5\end{array}\right).$

Exercise

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

$\left(\begin{array}{cc}5& 7\\ 3& 4\end{array}\right)$

over $ℂ$.

Exercise

Show that the characteristic polynomial of a linear mapping $\phi :V\to V$ 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

${\chi }_M(\lambda )=det(\lambda E_n-M)$

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

$M=\left(\begin{array}{ccccc}0& 1& 0& \dots & 0\\ 0& 0& 1& \dots & 0\\ ⋮& ⋮& \ddots & \ddots & ⋮\\ 0& 0& \dots & 0& 1\\ -a_0& -a_1& \dots & -a_{n-2}& -a_{n-1}\end{array}\right)$

equals

${\chi }_M=X^n+a_{n-1}{X}^{n-1}+\cdots +a_{1}X+a_0.$

Exercise

We consider the real matrix

$M=\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right).$

a) Determine

$M^n\left(\begin{array}{c}1\\ 0\end{array}\right)$

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

b) Let

$\left(\begin{array}{c}{x}_{n+1}\\ y_{n+1}\end{array}\right):=M^n\left(\begin{array}{c}1\\ 0\end{array}\right).$

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

$M=\left(\begin{array}{cccc}-1& 1& 0& 0\\ 0& -1& 1& 0\\ 0& 0& -1& 1\\ -1& 0& 0& 1\end{array}\right).$

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

$\left(\begin{array}{c}\frac{1}{(1+\lambda {)}^3}\\ \frac{1}{(1+\lambda {)}^2}\\ \frac{1}{(1+\lambda )}\\ 1\end{array}\right)$

is an eigenvector of the matrix

$\left(\begin{array}{cccc}-1& 1& 0& 0\\ 0& -1& 1& 0\\ 0& 0& -1& 1\\ -1& 0& 0& 1\end{array}\right)$

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

$\mathrm{\Psi }:{ℝ}_{\ge 0}^4⟶{ℝ}_{\ge 0}^4$

that assigns to a four tuple $(a,b,c,d)$ the four tuple

$(|b-a|,|c-b|,|d-c|,|a-d|).$

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

Exercise

Determine the eigenvalues and the eigenspaces of the linear mapping

$\phi :{ℝ}^3⟶{ℝ}^3,v⟼Mv,$

given by the matrix

$M=\left(\begin{array}{ccc}2& 0& 5\\ 0& -1& 0\\ 8& 0& 5\end{array}\right).$

Exercise

We consider the linear mapping

$\phi :{ℂ}^3⟶{ℂ}^3$

that is given by the matrix

$A=\left(\begin{array}{ccc}2& 1& -2+\mathrm{i}\\ 0& \mathrm{i}& 1+\mathrm{i}\\ 0& 0& -1+2\mathrm{i}\end{array}\right),$

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 $\phi$ with respect to a basis of eigenvectors.

Exercise

Let

$A=\left(\begin{array}{ccc}-4& 6& 6\\ 0& 2& 0\\ -3& 3& 5\end{array}\right)\in {\mathrm{Mat}}_{3\times 3}(ℝ).$

Compute:

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

Exercise

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

$\left(\begin{array}{ccc}2& 1& 3\\ 5& 0& 7\\ 9& 3& 8\end{array}\right).$

Exercise

Show that the matrix

$\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$

is diagonalizable over $\mathbb{Q}$.

Exercise

Let $M\in {\mathrm{Mat}}_n(K)$ 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 $1\le m\le n$. 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 $(5,-3)$.
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

$\left(\begin{array}{ccc}4& 7& -3\\ 2& 7& 5\\ 0& 0& -6\end{array}\right)$

is trigonalizable or not.

Exercise

Suppose that a linear mapping

$\phi :{ℝ}^2⟶{ℝ}^2$

is given by the matrix

$\left(\begin{array}{cc}3& 5\\ 0& 3\end{array}\right)$

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

$\left(\begin{array}{cc}3& 1\\ 0& 3\end{array}\right)$

with respect to this basis.

The next exercises use the following definition.

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

$\phi :V⟶V$

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

$\phi (U)\subseteq U$

holds.

Exercise

Let $\phi :V\to V$ 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 $\phi$-invariant.
2. $V$ is $\phi$-invariant.
3. Eigenspaces are $\phi$-invariant.
4. Let $U_1,U_2\subseteq V$ be $\phi$-invariant linear subspaces. Then also $U_1\cap U_2$ and $U_1+U_2$ are $\phi$-invariant.
5. Let $U\subseteq V$ be a $\phi$-invariant linear subspace. Then also the image space $\phi (U)$ and the preimage space ${\phi }^{-1}(U)$ are $\phi$-invariant.

Exercise

Let $\phi :V\to V$ a linear mapping on a $K$-vector space $V$ over a field $K$, and let $v\in V$. Show that the smallest $\phi$-invariant linear subspace of $V$ that contains $v$, equals

$⟨{\phi }^n(v),n\in \mathbb{N}⟩.$

Exercise

Let $\phi :V\to V$ a linear mapping on a $K$-vector space $V$ over a field $K$. Show that the subset of $V$, defined by

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

is an $\phi$-invariant linear subspace.

Exercise

Let $\phi :V\to V$ be a linear mapping on a $K$-vector space $V$. Let $v_1,\dots ,v_n$ be a basis of $V$, such that $\phi$ is described, with respect to this basis, by an upper triangular matrix. Show that the linear subspaces

$⟨v_1,\dots ,v_i⟩$

are $\phi$-invariant for every $i$.

Exercise

Determine, whether the real matrix

$\left(\begin{array}{cccc}-4& -1& -2& 3\\ 6& 7& 7& 1\\ 0& 0& 3& -2\\ 0& 0& 6& 2\end{array}\right)$

is trigonalizable or not.

Hand-in-exercises

Exercise (2 marks)

Compute the characteristic polynomial of the matrix

$\left(\begin{array}{ccc}-3& 8& 5\\ 4& 7& 1\\ 2& -4& 5\end{array}\right).$

Exercise (3 marks)

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

$\left(\begin{array}{cc}2& 7\\ 5& 4\end{array}\right)$

over $ℂ$.

Exercise (4 marks)

Let

$\phi :{ℝ}^3⟶{ℝ}^3$

be a linear mapping. Show that $\phi$ has at least one eigenvector.

Exercise (4 marks)

Let

$A=\left(\begin{array}{ccc}-5& 0& 7\\ 6& 2& -6\\ -4& 0& 6\end{array}\right)\in {\mathrm{Mat}}_{3\times 3}(ℝ).$

Compute:

1. the eigenvalues of $A$;
2. the corresponding eigenspaces;
3. the geometric and algebraic multiplicities of each eigenvalue;
4. a matrix $C\in {\mathrm{Mat}}_{3\times 3}(ℝ)$ 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

$M=\left(\begin{array}{ccc}3& -4& 5\\ 0& -1& 2\\ 0& 0& 3\end{array}\right).$

Exercise (4 marks)

Decide whether the matrix

$\left(\begin{array}{ccc}5& -1& 3\\ 7& 9& 8\\ 6& 2& -7\end{array}\right)$

is trigonalizable over $ℝ$.

Exercise (3 marks)

Determine whether the real matrix

$\left(\begin{array}{cccc}1& 5& 6& 2\\ 5& 7& -4& -3\\ 0& 0& -2& 8\\ 0& 0& 1& 9\end{array}\right)$

is trigonalizable or not.

Footnotes

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