Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 23

Exercise for the break

Exercise

Compute the characteristic polynomial of the matrix

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

Exercises

Exercise

Determine the characteristic polynomial and the eigenvalues of the linear mapping

$\phi :{ℝ}^3⟶{ℝ}^3,$

given by the matrix

$A=\left(\begin{array}{ccc}-2& -4& 2\\ 0& 2& 3\\ 0& 6& 1\end{array}\right)$

with respect to the standard 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

Let $K$ be a field, and let $M$ denote an $n\times n$-matrix over $K$. How can we find the $\mathrm{trace}\left(M\right)$ in the characteristic polynomial ${\chi }_M$?

Exercise

Determine the characteristic polynomial of a matrix

$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right).$

What is the relevance of the coefficients of this polynomial?

Exercise

Determine the characteristic polynomial of a matrix

$\left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right).$

What is the relevance of the coefficients of this polynomial?

Exercise

Compute the characteristic polynomial of the matrix

$\left(\begin{array}{cc}\frac{X^2-4X+7}{X-1}& \frac{6X-5}{X^2-5}\\ \frac{4X^2+3X-2}{6X^2-3X}& \frac{X^2+8X+9}{X^2-3X+5}\end{array}\right)$

over the field of rational functions $\mathbb{Q}(X)$.

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

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

$M=\left(\begin{array}{cc}-\frac{1}{2}& \frac{1}{4}\\ -3& -\frac{1}{2}\end{array}\right)$

over $ℂ$.

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

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, besides the preceding exercises also Exercise 10.16 is helpful.

Exercise

We consider the mapping

$\Psi :{ℝ}_{\ge 0}^4⟶{ℝ}_{\ge 0}^4$

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

Let $K$ be a field, and let $M$ denote an $n\times n$-matrix over $K$ with the property, that the characteristic polynomial splits into linear factors, that is,

${\chi }_M=(X-{\lambda }_1{)}^{{\mu }_1}\cdot (X-{\lambda }_2{)}^{{\mu }_2}\cdots (X-{\lambda }_k{)}^{{\mu }_k}.$

Show that

$\mathrm{trace}\left(M\right)=\sum _{i=1}^k{\mu }_i{\lambda }_i.$

Exercise

Let $K$ be the field with two elements, we consider the matrix

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

over $K$. Show that the characteristic polynomial ${\chi }_M$ is not the zero polynomial, but that

${\chi }_M(\lambda )=0$

holds for all $\lambda \in K$.

Exercise

Show that a square matrix and its transposed matrix have the same characteristic polynomial.

Exercise

What is wrong in the following argumentation:

"For two $n\times n$-matrices $M,N$, the characteristic polynomials fulfill the relation

${\chi }_{M\circ N}={\chi }_M{\chi }_N.$

This is because, by definition, we have

${\chi }_{M\circ N}=det(X{E}_n-M\circ N)=det(X{E}_n-M)det(X{E}_n-N)={\chi }_M\cdot {\chi }_N,$

where the equation in the middle rests on the multiplication theorem for determinants“.

Exercise

Let $M$ be an $n\times n$-matrix, with the characteristic polynomial

${\chi }_M=X^n+c_{n-1}{X}^{n-1}+c_{n-2}{X}^{n-2}+\cdots +c_2{X}^2+c_{1}X+c_0.$

Determine the characteristic polynomial of the scaled matrix $sM$, $s\in K$.

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

Let $K\subseteq L$ be a field extension. Let an $n\times n$-matrix $M$ over $K$ be given. Show that the characteristic polynomial ${\chi }_M\in K[X]$ coincides with the characteristic polynomial of $M$, considered as a matrix over $L$.

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 $V$ be a finite-dimensional $K$-vector space, and let $\phi \in {\mathrm{End}}_{}\left(V\right)$. Show that the following statements are equivalent:

1. The linear mapping $\phi$ is an isomorphism.
2. $0$ is not an eigenvalue of $\phi$.
3. The constant term of the characteristic polynomial ${\chi }_{\phi }$ is $\ne 0$.

Exercise

Let

$\phi :V⟶V$

be an endomorphism on a finite-dimensional $K$-vector space $V$, and let $a\in K$ an eigenvalue of $\phi$. Show that $a$ is also an eigenvalue of the dual mapping

${\phi }^{\ast }:V^{\ast }⟶V^{\ast }.$

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 $K$ denote a field, and let $V$ denote a $K$-vector space of finite dimension. Let

$\phi :V⟶V$

be a linear mapping. Suppose that the characteristic polynomial ${\chi }_{\phi }$ factors into different linear factors. Show that $\phi$ is diagonalizable.

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 $K$ be a field, let $V$ be a $K$-vector space, and let

$\phi :V⟶V$

a linear mapping. Let $U\subseteq V$ be a $\phi$-invariant linear subspace of $V$. Show that, for a polynomial $P\in K[X]$, the space $U$ is also $P(\phi )$-invariant.

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

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 finite-dimensional $K$-vector space $V$. Let $k\le n$. Show that there exists an invariant linear subspace $U\subseteq V$ of dimension $k$, if and only if there exists a basis of $V$ such that the describing matrix of $\phi$, with respect to this basis, has the form

$\left(\begin{array}{cccccc}{a}_{11}& \dots & a_{1k}& a_{1k+1}& \dots & a_{1n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{k1}& \dots & a_{kk}& a_{kk+1}& \dots & a_{kn}\\ 0& \dots & 0& a_{k+1k+1}& \dots & a_{k+1n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& \dots & 0& a_{nk+1}& \dots & a_{nn}\end{array}\right).$

Exercise

Let $\phi :V\to V$ be a linear mapping on the finite-dimensional $K$-vector space $V$. Let $k\le n$. Show that there exists a direct sum decomposition $V=U\oplus W$ into invariant linear subspaces $U,W\subseteq V$ of dimension $k$ and $n-k$, if and only if there exists a basis of $V$ such that the describing matrix of $\phi$ with respect to this basis has the form

$\left(\begin{array}{cccccc}{a}_{11}& \dots & a_{1k}& 0& \dots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{k1}& \dots & a_{kk}& 0& \dots & 0\\ 0& \dots & 0& a_{k+1k+1}& \dots & a_{k+1n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& \dots & 0& a_{nk+1}& \dots & a_{nn}\end{array}\right).$

Exercise

Let $V$ be a finite-dimensional $K$-vector space, and $U\subseteq V$ a linear subspace. Show that

$R(U)=\{\phi \in {\mathrm{End}}_{}\left(V\right)\mid U\text{ is a }\phi -\text{invariant linear subspace}\}$

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

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

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

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)

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 (4 marks)

Let

$\phi :{ℝ}^3⟶{ℝ}^3$

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

Footnotes

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