Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 28

Exercise for the break

Exercise

We consider the matrix

${\begin{pmatrix}2&3\\0&5\end{pmatrix}},$

and the three decompositions

${}{\begin{pmatrix}2&3\\0&5\end{pmatrix}}={\begin{pmatrix}2&0\\0&5\end{pmatrix}}+{\begin{pmatrix}0&3\\0&0\end{pmatrix}}\,,$

${}{\begin{pmatrix}2&3\\0&5\end{pmatrix}}={\begin{pmatrix}2&3\\0&5\end{pmatrix}}+{\begin{pmatrix}0&0\\0&0\end{pmatrix}}\,$

and

${}{\begin{pmatrix}2&0\\0&5\end{pmatrix}}={\begin{pmatrix}2&0\\0&5\end{pmatrix}}+{\begin{pmatrix}0&0\\0&0\end{pmatrix}}\,.$

Which one is (or are) the canonical additive decomposition in the sense of Theorem 28.1 (and with respect to which basis)?

Exercises

Exercise

Let

${}D={\begin{pmatrix}a_{11}&0&\cdots &\cdots &0\\0&a_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&a_{n-1\,n-1}&0\\0&\cdots &\cdots &0&a_{nn}\end{pmatrix}}\,.$

Show that ${}D$ commutes with all ${}n\times n$-matrices ${}M$ if and only if all its diagonal entries coincide.

Exercise

Describe the direct sum decomposition of the ${}p_{i}$ with respect to the generalized eigenspaces in the proof of Theorem 28.1 .

Exercise

Suppose that a linear mapping

$\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$

is given by the matrix

${\begin{pmatrix}3&5\\0&3\end{pmatrix}}$

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

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

with respect to this basis.

Exercise

Suppose that the linear mapping

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

is given by the matrix

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

with respect to the standard basis. Find a basis such that ${}\varphi$ is described by the matrix

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

with respect to the new basis.

Exercise

Determine, for the real matrix

${}M={\begin{pmatrix}3&5&2&4\\0&3&6&-1\\0&0&3&2\\0&0&0&3\end{pmatrix}}\,,$

its Jordan normal form (we do not ask for a basis).

Exercise

Let ${}M$ be a nilpotent ${}n\times n$-Jordan matrix. Show, that the kernels ${}\operatorname {kern} M^{i}$ form a flag in ${}K^{n}$.

Exercise

Let ${}M$ be an ${}n\times n$-Jordan matrix to the eigenvalue ${}\lambda$. Determine the minimal polynomial of ${}M$.

Exercise

Let ${}M$ be an ${}n\times n$-matrix with Jordan blocks ${}J_{1},\ldots ,J_{k}$, and suppose that all diagonal entries are ${}\lambda$. Determine the minimal polynomial of ${}M$.

Exercise

Let ${}M$ be a matrix in Jordan normal form, where only one eigenvalue occurs. Show that the number of the Jordan blocks in ${}M$ equals the dimension of the eigenspace.

Exercise

Show that the product of two matrices in Jordan normal form is, in general, not in Jordan normal form.

Exercise

Let ${}M$ be an ${}n\times n$-matrix in Jordan normal form, and let ${}D$ denote the corresponding diagonal matrix. Show that the canonical additive decomposition in the sense of Theorem 28.1 is

${}M=D+(M-D)\,.$

Exercise

Let ${}\varphi \colon V\rightarrow V$ be a trigonalizable linear mapping, together with its canonical additive decomposition

${}\varphi =\varphi _{\rm {diag}}+\varphi _{\rm {nil}}\,$

in the sense of Theorem 28.1 . Show that the eigenvalues of ${}\varphi$ and of ${}\varphi _{\rm {diag}}$ coincide. Does this also hold for their algebraic multiplicity? Does this also hold for their geometric multiplicity?

Exercise *

Let

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

with ${}\lambda \in K$. Show by induction that

${}M^{n}={\begin{pmatrix}\lambda ^{n}&n\lambda ^{n-1}\\0&\lambda ^{n}\end{pmatrix}}\,$

holds.

Exercise

The matrix

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

describes a quarter-turn in the real plane. Diagonalize this matrix over the complex numbers.

Exercise

Let ${}\varphi \colon V\rightarrow V$ be a linear mapping on a finite-dimensional ${}K$-vector space ${}V$. Show that ${}\varphi$ has finite order if and only if the minimal polynomial of ${}\varphi$ is a divisor of ${}X^{n}-1$ for some ${}n\in \mathbb {N} _{+}$.

Exercise

Let ${}\varphi \colon V\rightarrow V$ be a linear mapping on a finite-dimensional ${}\mathbb {C}$-vector space ${}V$, and suppose that ${}\varphi$ has finite order. Show that the characteristic polynomial of ${}\varphi$ divides a polynomial of the form ${}{\left(X^{n_{1}}-1\right)}\cdots {\left(X^{n_{r}}-1\right)}$. Show also that the characteristic polynomial does not, in general, divide a polynomial of the form ${}X^{n}-1$.

Exercise

Let ${}\varphi \colon V\rightarrow V$ be a linear mapping on a finite-dimensional ${}\mathbb {R}$-vector space ${}V$, and suppose that ${}\varphi$ has finite order. Show that the characteristic polynomial of ${}\varphi$ divides a polynomial of the form ${}{\left(X^{n_{1}}-1\right)}\cdots {\left(X^{n_{r}}-1\right)}$.

We say that a field ${}K$ has positive characteristic, if there exists a positive natural number ${}p\in \mathbb {N} _{+}$ such that the equation ${}p=0$ holds. The fields ${}\mathbb {Q} ,\mathbb {R} ,\mathbb {C}$ do not have this property, we say that there characteristic is ${}0$. Finite fields have positive characteristic, their characteristic is always a prime number.

Exercise

Let ${}K$ be a field of positive characteristic ${}p>0$. Show that the matrix

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

has finite order ${}p$.

Hand-in-exercises

Exercise (3 marks)

We consider the linear mapping

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

that is described, with respect to the standard basis, by the matrix

${\begin{pmatrix}2&3&4\\0&2&5\\0&0&2\end{pmatrix}}.$

Find a basis such that ${}\varphi$ is described, with respect to this basis, by the matrix

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

Exercise (4 marks)

Determine a basis such that linear mapping given by

${}M={\begin{pmatrix}4&-6&0&-1\\0&5&2&7\\0&0&4&4\\0&0&0&5\end{pmatrix}}\,$

is in Jordan normal form with respect to this basis.

Exercise (2 marks)

Let ${}M$ be a Jordan matrix to the eigenvalue ${}\lambda$. Show that the eigenspace of ${}M$ to the eigenvalue ${}\lambda$ is one-dimensional, and that there exist no further eigenvectors.

Exercise (4 marks)

Let

$\varphi \colon V\longrightarrow V$

be an endomorphism that is described, with respect to a suitable basis, by a ${}n\times n$-Jordan matrix. Show that there does not exist a direct sum decomposition

${}V=U\oplus W\,$

into ${}\varphi$-invariant linear subspaces ${}U,W\subset V$.

Exercise (8 (4+2+2) marks)

Let ${}n\in \mathbb {N} _{+}$, and let a field ${}K$ of characteristic ${}0$ be fixed. For a nilpotent ${}n\times n$-matrix ${}M$, let ${}\exp M$ be defined by

${}\exp M=\sum _{k=0}^{n}{\frac {1}{k!}}M^{k}\,.$

a) Show that for commuting nilpotent matrices ${}M,N$, the equality

${}\exp {\left(M+N\right)}=\exp M\circ \exp N\,$

holds.

b) Show that for a nilpotent matrix ${}M$, the matrix ${}\exp M$ is invertible.

c) Show that for a nilpotent matrix ${}M$, the matrix ${}\exp M$ is unipotent.

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