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

Exercise for the break

Exercise

Compute the characteristic polynomial of the matrix

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

Exercises

Exercise

Determine the characteristic polynomial and the eigenvalues of the linear mapping

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

given by the matrix

${}A={\begin{pmatrix}-2&-4&2\\0&2&3\\0&6&1\end{pmatrix}}\,$

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

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

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

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

Exercise

Determine the characteristic polynomial of a matrix

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}.$

What is the relevance of the coefficients of this polynomial?

Exercise

Determine the characteristic polynomial of a matrix

${\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}.$

What is the relevance of the coefficients of this polynomial?

Exercise

Compute the characteristic polynomial of the matrix

${\begin{pmatrix}{\frac {X^{2}-4X+7}{X-1}}&{\frac {6X-5}{X^{2}-5}}\\{\frac {4X^{2}+3X-2}{6X^{2}-3X}}&{\frac {X^{2}+8X+9}{X^{2}-3X+5}}\end{pmatrix}}$

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

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

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

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

over ${}\mathbb {C}$.

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

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

that 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

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, besides the preceding exercises also Exercise 10.16 is helpful.

Exercise

We consider the mapping

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

that 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 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

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

Exercise

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

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

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

${}\chi _{M\circ N}=\det {\left(XE_{n}-M\circ N\right)}=\det {\left(XE_{n}-M\right)}\det {\left(XE_{n}-N\right)}=\chi _{M}\cdot \chi _{N}\,,$

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 ${}1\leq m\leq n$. 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

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

Exercise

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

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

Exercise

Let

$\varphi \colon V\longrightarrow V$

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

${\varphi }^{*}\colon {V}^{*}\longrightarrow {V}^{*}.$

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

$\varphi \colon V\longrightarrow V$

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

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 ${}V$ that contains ${}v$, equals

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

Exercise

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

$\varphi \colon V\longrightarrow V$

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

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

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 finite-dimensional ${}K$-vector space ${}V$. Let ${}k\leq n$. 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 ${}\varphi$, with respect to this basis, has the form

${\begin{pmatrix}a_{11}&\ldots &a_{1k}&a_{1k+1}&\ldots &a_{1n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\a_{k1}&\ldots &a_{kk}&a_{kk+1}&\ldots &a_{kn}\\0&\ldots &0&a_{k+1k+1}&\ldots &a_{k+1n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\0&\ldots &0&a_{nk+1}&\ldots &a_{nn}\end{pmatrix}}.$

Exercise

Let ${}\varphi \colon V\rightarrow V$ be a linear mapping on the finite-dimensional ${}K$-vector space ${}V$. Let ${}k\leq n$. 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 ${}\varphi$ with respect to this basis has the form

${\begin{pmatrix}a_{11}&\ldots &a_{1k}&0&\ldots &0\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\a_{k1}&\ldots &a_{kk}&0&\ldots &0\\0&\ldots &0&a_{k+1k+1}&\ldots &a_{k+1n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\0&\ldots &0&a_{nk+1}&\ldots &a_{nn}\end{pmatrix}}.$

Exercise

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

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

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

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

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

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)

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 (4 marks)

Let

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

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

Footnotes

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