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

Exercise for the break

Exercise

We consider the matrix

$\left(\begin{array}{cc}2& 3\\ 0& 5\end{array}\right),$

and the three decompositions

$\left(\begin{array}{cc}2& 3\\ 0& 5\end{array}\right)=\left(\begin{array}{cc}2& 0\\ 0& 5\end{array}\right)+\left(\begin{array}{cc}0& 3\\ 0& 0\end{array}\right),$

$\left(\begin{array}{cc}2& 3\\ 0& 5\end{array}\right)=\left(\begin{array}{cc}2& 3\\ 0& 5\end{array}\right)+\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$

and

$\left(\begin{array}{cc}2& 0\\ 0& 5\end{array}\right)=\left(\begin{array}{cc}2& 0\\ 0& 5\end{array}\right)+\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right).$

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

$D=\left(\begin{array}{ccccc}{a}_{11}& 0& \cdots & \cdots & 0\\ 0& a_{22}& 0& \cdots & 0\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ 0& \cdots & 0& a_{n-1n-1}& 0\\ 0& \cdots & \cdots & 0& a_{nn}\end{array}\right).$

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

$\phi :{ℝ}^2⟶{ℝ}^2$

is given by the matrix

$\left(\begin{array}{cc}3& 5\\ 0& 3\end{array}\right)$

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

$\left(\begin{array}{cc}3& 1\\ 0& 3\end{array}\right)$

with respect to this basis.

Exercise

Suppose that the linear mapping

$\phi :{ℝ}^3⟶{ℝ}^3$

is given by the matrix

$\left(\begin{array}{ccc}-2& 2& 7\\ 0& -2& -6\\ 0& 0& -2\end{array}\right)$

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

$\left(\begin{array}{ccc}-2& 1& 0\\ 0& -2& 1\\ 0& 0& -2\end{array}\right)$

with respect to the new basis.

Exercise

Determine, for the real matrix

$M=\left(\begin{array}{cccc}3& 5& 2& 4\\ 0& 3& 6& -1\\ 0& 0& 3& 2\\ 0& 0& 0& 3\end{array}\right),$

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 $\mathrm{kern}{M}^i$ 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 $J_1,\dots ,J_k$, 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 $\phi :V\to V$ be a trigonalizable linear mapping, together with its canonical additive decomposition

$\phi ={\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}+{\phi }_{\mathrm{n}\mathrm{i}\mathrm{l}}$

in the sense of Theorem 28.1 . Show that the eigenvalues of $\phi$ and of ${\phi }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}$ coincide. Does this also hold for their algebraic multiplicity? Does this also hold for their geometric multiplicity?

Exercise *

Let

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

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

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

holds.

Exercise

The matrix

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

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

Exercise

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

Exercise

Let $\phi :V\to V$ be a linear mapping on a finite-dimensional $ℂ$-vector space $V$, and suppose that $\phi$ has finite order. Show that the characteristic polynomial of $\phi$ divides a polynomial of the form $(X^{n_1}-1)\cdots (X^{n_r}-1)$. Show also that the characteristic polynomial does not, in general, divide a polynomial of the form $X^n-1$.

Exercise

Let $\phi :V\to V$ be a linear mapping on a finite-dimensional $ℝ$-vector space $V$, and suppose that $\phi$ has finite order. Show that the characteristic polynomial of $\phi$ divides a polynomial of the form $(X^{n_1}-1)\cdots (X^{n_r}-1)$.

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 $\mathbb{Q},ℝ,ℂ$ 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

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

has finite order $p$.

Hand-in-exercises

Exercise (3 marks)

We consider the linear mapping

$\phi :{ℝ}^3⟶{ℝ}^3$

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

$\left(\begin{array}{ccc}2& 3& 4\\ 0& 2& 5\\ 0& 0& 2\end{array}\right).$

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

$\left(\begin{array}{ccc}2& 1& 0\\ 0& 2& 1\\ 0& 0& 2\end{array}\right).$

Exercise (4 marks)

Determine a basis such that linear mapping given by

$M=\left(\begin{array}{cccc}4& -6& 0& -1\\ 0& 5& 2& 7\\ 0& 0& 4& 4\\ 0& 0& 0& 5\end{array}\right)$

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

$\phi :V⟶V$

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 $\phi$-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

$\exp (M+N)=\exp M\circ \exp N$

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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