Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 24

Exercise for the break

Exercise

Confirm the Theorem of Cayley-Hamilton for the matrix

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

by an explicit computation.

Exercises

Exercise

Confirm the Theorem of Cayley-Hamilton for the matrix

${\begin{pmatrix}3&1&0&0\\0&3&0&0\\0&0&2&1\\0&0&0&2\end{pmatrix}}$

by an explicit computation.

Exercise

Confirm the Theorem of Cayley-Hamilton for an upper triangular matrix of the form

${\begin{pmatrix}a&b&c\\0&a&d\\0&0&a\end{pmatrix}}.$

Exercise

Let ${}M$ be a diagonalizable matrix with the characteristic polynomial ${}\chi _{M}$. Show directly that

${}\chi _{M}\,(M)=0\,$

holds.

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. Let

$\psi =\varphi \oplus \cdots \oplus \varphi \colon V\oplus \cdots \oplus V\longrightarrow V\oplus \cdots \oplus V$

be the ${}m$-fold direct sum of ${}\varphi$ with itself. How is the minimal polynomial (the characteristic polynomial) of ${}\psi$ related to the minimal polynomial (the characteristic polynomial) of ${}\varphi$?

Exercise

Express the matrix

${}M={\begin{pmatrix}4X^{2}-3X+2&X^{3}-2X+8\\3X^{4}-X^{3}-2X^{2}+7&X^{4}-6\end{pmatrix}}\,$

(whose entries are in ${}\mathbb {Q} [X]\subset \mathbb {Q} (X)$) in the form

$A_{4}X^{4}+A_{3}X^{3}+A_{2}X^{2}+A_{1}X+A_{0}$

with matrices ${}A_{4},A_{3},A_{2},A_{1},A_{0}\in \operatorname {Mat} _{2}(\mathbb {Q} )$.

Exercise

Let ${}M$ be an ${}n\times n$-matrix over a field ${}K$, and suppose that its minimal polynomial has the form

$(X-\lambda _{1})\cdots (X-\lambda _{k})$

with different ${}\lambda _{i}$. Show that ${}M$ is diagonalizable.

Exercise

Let ${}K$ be a field, let ${}n\in \mathbb {N}$, and let ${}M$ be the set of all ${}n$-th roots of unity in ${}K$. Show that ${}M$ is a subgroup of the unit group ${}K^{\times }$.

Exercise

Show that every complex root of unity lies on the unit circle.

An ${}n$-th root of unity is called primitive, if its order

is ${}n$.

Exercise

Let ${}\zeta \in K$ be an ${}n$-th primitive root of unity in a field ${}K$. Show the formula

${}1+\zeta +\zeta ^{2}+\cdots +\zeta ^{n-1}=0\,.$

Exercise

Let ${}M$ be the permutation matrix for a transposition. Show that ${}M$ is diagonalizable over ${}\mathbb {R}$.

Exercise

Let the cycle ${}1\mapsto 2\mapsto 3\mapsto \ldots \mapsto n\mapsto 1$ be given, and let ${}M$ denote the corresponding ${}n\times n$-permutation matrix over a field ${}K$.

a) Let ${}P\in K[X]$ be a polynomial of degree ${}<n$. Establish a formula for ${}(P(M))(e_{1})$.

b) Determine the minimal polynomial of ${}M$.

c) Give an example for an endomorphism ${}\varphi$ on a real vector space ${}V$ with different vectors ${}v_{1},v_{2},v_{3}\in V$ such that ${}\varphi (v_{1})=v_{2}$, ${}\varphi (v_{2})=v_{3}$ and ${}\varphi (v_{3})=v_{1}$ holds, and such that the minimal polynomial of ${}\varphi$ is not ${}X^{3}-1$.

Exercise

Suppose that, for some permutation ${}\pi \in S_{n}$, its cycle decomposition is known. Determine the minimal polynomial and the characteristic polynomial of the permutation matrix ${}M_{\pi }$.

Exercise

Let ${}\pi \in S_{n}$ be a permutation, and let ${}M_{\pi }$ denote the corresponding permutation matrix over a field ${}K$. For ${}J\subseteq \{1,\ldots ,n\}$, let

${}V_{J}=\langle e_{j},\,j\in J\rangle \subseteq K^{n}\,.$

a) Show that ${}V_{J}$ is ${}M_{\pi }$-invariant if and only if ${}\pi (J)\subseteq J$.

b) Show that there might exist ${}M_{\pi }$-invariant subspaces that are not of the form ${}V_{J}$.

Exercise

Let ${}K$ be a finite field. Show that every unit in ${}K$ is a root of unity.

Exercise *

Determine the order of the matrix

${\begin{pmatrix}0&1\\2&0\end{pmatrix}}$

over the field with ${}3$ elements.

Exercise

Let ${}K$ be a finite field, and let ${}M$ denote an invertible ${}n\times n$-matrix over ${}K$. Show that ${}M$ has finite order.

Exercise

Give a matrix ${}M\in \operatorname {GL} _{2}\!{\left(\mathbb {Q} \right)}$ of order ${}4$.

Hand-in-exercises

Exercise (4 marks)

Confirm the Theorem of Cayley-Hamilton for the matrix

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

by an explicit computation.

Exercise (4 marks)

Let ${}M$ be an ${}n\times n$-matrix over a field ${}K$, and let

${}P=a_{0}+a_{1}X+\cdots +a_{m}X^{m}\in K[X]\,$

be a polynomial with

${}P(M)=0\,$

and with ${}a_{0}\neq 0$. Show that ${}M$ is invertible, and that its inverse matrix is given by

${}M^{-1}=-{\frac {1}{a_{0}}}{\left(a_{1}+a_{2}M+\cdots +a_{m}M^{m-1}\right)}\,.$

Exercise (5 marks)

Let ${}V$ and ${}W$ be finite-dimensional ${}K$-vector spaces, and let

$\varphi \colon V\longrightarrow V$

and

$\psi \colon W\longrightarrow W$

be endomorphisms, with the minimal polynomials ${}P$ and ${}Q$. Show that the minimal polynomial of

$\varphi \oplus \psi \colon V\oplus W\longrightarrow V\oplus W$

equals the normed generator of the ideal ${}(P)\cap (Q)$.

Exercise (3 marks)

Determine the order of the matrix

${\begin{pmatrix}4&1\\2&3\end{pmatrix}}$

over the field ${}\mathbb {Z} /(5)$ with ${}5$ elements.

Exercise (4 marks)

Show that a permutation matrix over ${}\mathbb {C}$ is diagonalizable.

<< | Linear algebra (Osnabrück 2024-2025)/Part I | >> PDF-version of this exercise sheet Lecture for this exercise sheet (PDF)