Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 25

Exercise for the break

Exercise

Show that the matrix

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

is trigonalizable.

Exercises

Exercise

Determine, whether the real matrix

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

is trigonalizable or not.

Exercise

Show that the matrix

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

is not trigonalizable over ${}\mathbb {R}$.

Exercise

Determine whether the matrix

${\begin{pmatrix}4&2&2\\1&1&3\\3&0&4\end{pmatrix}}$

over the field with five elements is trigonalizable or not.

Exercise

Let ${}V_{1},\ldots ,V_{n}$ denote finite-dimensional vector spaces over the field ${}K$, let

$\varphi _{i}\colon V_{i}\longrightarrow V_{i}$

denote linear mappings, and let

$\varphi =\varphi _{1}\times \cdots \times \varphi _{n}\colon V_{1}\times \cdots \times V_{n}\longrightarrow V_{1}\times \cdots \times V_{n}$

denote the product mapping. Show that ${}\varphi$ is trigonalizable if and only if this holds for all ${}\varphi _{i}$.

Exercise

Let ${}\varphi \colon V\rightarrow V$ be a trigonalizable endomorphism on the finite-dimensional ${}K$-vector space ${}V$, and let ${}P\in K[X]$ be a polynomial. Show that ${}P(\varphi )$ is also trigonalizable.

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, and let

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

denote its dual mapping. Show that ${}\varphi$ is trigonalizable if and only if ${}{\varphi }^{*}$ is trigonalizable.

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. Show that ${}\varphi$ is trigonalizable if and only if ${}\varphi$ is described by a lower triangular matrix

${\begin{pmatrix}a_{1}&0&\cdots &\cdots &0\\\ast &a_{2}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\\ast &\cdots &\ast &a_{n-1}&0\\\ast &\cdots &\cdots &\ast &a_{n}\end{pmatrix}}$

with respect to a suitable basis.

Exercise

Let ${}M$ be a ${}2\times 2$-matrix over a field ${}K$. Show that ${}M$ is trigonalizable if and only if ${}M$ has an eigenvector.

Exercise

Show that the composition of two diagonalizable mappings is, in general, not trigonalizable.

Exercise

Determine whether the permutation matrix

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

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

Exercise

Determine the minimal polynomials of the left-upper submatrices of

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

Exercise

Determine whether the following chain of linear subspaces in ${}\mathbb {R} ^{3}$ is a flag.

$V_{0}=0,\,V_{1}=\mathbb {R} {\begin{pmatrix}2\\7\\-5\end{pmatrix}},\,V_{2}=\langle {\begin{pmatrix}4\\-3\\9\end{pmatrix}},{\begin{pmatrix}-6\\-4\\4\end{pmatrix}}\rangle ,\,V_{3}=\mathbb {R} ^{3}.$

Exercise

Determine whether the following chain of linear subspaces in ${}\mathbb {R} ^{3}$ is a flag.

$V_{0}=0,\,V_{1}=\mathbb {R} {\begin{pmatrix}8\\-3\\1\end{pmatrix}},$

$V_{2}=\operatorname {kern} \varphi ,{\text{ where }}\varphi :\mathbb {R} ^{3}\rightarrow \mathbb {R} {\text{ is given by the matrix}}\left(1,\,2,\,-2\right),\,V_{3}=\mathbb {R} ^{3}.$

Exercise

Let ${}V$ be a finite-dimensional ${}K$-vector space. Show that there exist flags in ${}V$.

Exercise

Let ${}V$ and ${}W$ be vector spaces over ${}K$ of the same dimension ${}n$, and let

${}0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$

and

${}0=W_{0}\subset W_{1}\subset \ldots \subset W_{n-1}\subset W_{n}=W\,$

be flags in ${}V$ and in ${}W$, respectively. Show that there exists a bijective linear mapping

$\varphi \colon V\longrightarrow W$

such that

${}\varphi (V_{i})=W_{i}\,$

for all ${}i=0,1,\ldots ,n$.

Exercise

Let ${}K$ be a finite field with ${}q$ elements, and let ${}V$ denote a two-dimensional ${}K$-vector space. Determine the number of flags in ${}V$.

Exercise

Let ${}K$ be the field with three elements, and let ${}V$ denote a three-dimensional ${}K$-vector space. Determine the number of flags in ${}V$.

Exercise

Let ${}V$ be an ${}n$-dimensional ${}K$-vector space over a field ${}K$, and let

${}0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$

denote a flag in ${}V$. Show that the linear subspaces

${}W_{i}:=V_{n-i}^{\perp }\,$

define a flag in the dual space ${}{V}^{*}$.

Exercise

Let

${}0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$

be a flag in a ${}\mathbb {C}$-vector space ${}V$. We consider ${}V$ as a real vector space of real dimension ${}2n$. Show that there exist real linear subspaces

${}W_{i}\subseteq V\,$

such that

${}0\subset W_{1}\subset V_{1}\subset W_{2}\subset V_{2}\subset \ldots \subset V_{n-1}\subset W_{n}\subset V_{n}\,$

is a real flag.

Exercise

Let ${}V$ be an ${}n$-dimensional ${}K$-vector space over a field ${}K$. Let

${}0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$

be a flag in ${}V$. Show that there exists a bijective linear mapping

$\varphi \colon V\longrightarrow V$

such that this flag is the only ${}\varphi$-invariant flag.

Exercise

Let

${}M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\,$

be a matrix over a field ${}K$.

a) Show that there exists a matrix that is similar to ${}M$, and where at least one entry equals ${}0$.

b) Show that, in general, there does not exist a matrix that is similar to ${}M$, and where at least two entries equal ${}0$.

Hand-in-exercises

Exercise (4 marks)

Trigonalize the complex matrix

${\begin{pmatrix}2+{\mathrm {i} }&3\\5{\mathrm {i} }&1-{\mathrm {i} }\end{pmatrix}}.$

Exercise (4 marks)

Decide whether the matrix

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

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

Exercise (3 marks)

Determine whether the real matrix

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

is trigonalizable or not.

Exercise (4 marks)

Let ${}M$ be a real matrix that is over ${}\mathbb {R}$ not trigonalizable. Show that ${}M$ is over ${}\mathbb {C}$ diagonalizable.

Exercise (4 marks)

Let

${}M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}\,$

be a matrix over ${}\mathbb {Q}$, whose trace is ${}0$. Show that there exists a matrix ${}N$ of the form

${}N={\begin{pmatrix}0&r\\s&0\end{pmatrix}}\,$

that is similar to ${}M$.

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