## All the Conjugacy Classes of the Dihedral Group $D_8$ of Order 8

## Problem 54

Determine all the conjugacy classes of the dihedral group

\[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle\]
of order $8$.

of the day

Determine all the conjugacy classes of the dihedral group

\[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle\]
of order $8$.

Let $D_8$ be the dihedral group of order $8$.

Using the generators and relations, we have

\[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle.\]

**(a)** Let $A$ be the subgroup of $D_8$ generated by $r$, that is, $A=\{1,r,r^2,r^3\}$.

Prove that the centralizer $C_{D_8}(A)=A$.

**(b)** Show that the normalizer $N_{D_8}(A)=D_8$.

**(c) **Show that the center $Z(D_8)=\langle r^2 \rangle=\{1,r^2\}$, the subgroup generated by $r^2$.

Let $n$ be a positive integer. Let $D_{2n}$ be the dihedral group of order $2n$. Using the generators and the relations, the dihedral group $D_{2n}$ is given by

\[D_{2n}=\langle r,s \mid r^n=s^2=1, sr=r^{-1}s\rangle.\]
Put $\theta=2 \pi/n$.

\cos \theta & -\sin \theta\\

\sin \theta& \cos \theta

\end{bmatrix}$ is the matrix representation of the linear transformation $T$ which rotates the $x$-$y$ plane about the origin in a counterclockwise direction by $\theta$ radians.

\[ \rho(r)=\begin{bmatrix}

\cos \theta & -\sin \theta\\

\sin \theta& \cos \theta

\end{bmatrix} \text{ and }

\rho(s)=\begin{bmatrix}

0 & 1\\

1& 0

\end{bmatrix}\] extends to a homomorphism of $D_{2n}$ into $\GL_2(\R)$.

Let $A$ and $B$ be an $n \times n$ matrices.

Suppose that all the eigenvalues of $A$ are distinct and the matrices $A$ and $B$ commute, that is $AB=BA$.

Then prove that each eigenvector of $A$ is an eigenvector of $B$.

(It could be that each eigenvector is an eigenvector for distinct eigenvalues.)

Add to solve laterLet

\[A=\begin{bmatrix}

\frac{1}{7} & \frac{3}{7} & \frac{3}{7} \\

\frac{3}{7} &\frac{1}{7} &\frac{3}{7} \\

\frac{3}{7} & \frac{3}{7} & \frac{1}{7}

\end{bmatrix}\]
be $3 \times 3$ matrix. Find

\[\lim_{n \to \infty} A^n.\]

(*Nagoya University Linear Algebra Exam*)

Let $A$ and $B$ be normal subgroups of a group $G$. Suppose $A\cap B=\{e\}$, where $e$ is the unit element of the group $G$.

Show that for any $a \in A$ and $b \in B$ we have $ab=ba$.

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

Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ are linearly independent vectors in $V$.

Are the following vectors linearly independent?

\[\mathbf{v}_1+\mathbf{v}_2, \quad \mathbf{v}_2+\mathbf{v}_3, \quad \dots, \quad \mathbf{v}_{k-1}+\mathbf{v}_k, \quad \mathbf{v}_k+\mathbf{v}_1.\]

If it is linearly dependent, give a non-trivial linear combination of these vectors summing up to the zero vector.

Add to solve laterLet $T=\begin{bmatrix}

1 & 0 & 2 \\

0 &1 &1 \\

0 & 0 & 2

\end{bmatrix}$.

Calculate and simplify the expression

\[-T^3+4T^2+5T-2I,\]
where $I$ is the $3\times 3$ identity matrix.

(*The Ohio State University Linear Algebra Exam*)

Let $A$ be an $n\times n$ matrix such that $A^k=I_n$, where $k\in \N$ and $I_n$ is the $n \times n$ identity matrix.

Show that the trace of $(A^{-1})^{\trans}$ is the conjugate of the trace of $A$. That is, show that $\tr((A^{-1})^{\trans})=\overline{\tr(A)}$.

Add to solve later

Calculate the determinants of the following $n\times n$ matrices.

\[A=\begin{bmatrix}

1 & 0 & 0 & \dots & 0 & 0 &1 \\

1 & 1 & 0 & \dots & 0 & 0 & 0 \\

0 & 1 & 1 & \dots & 0 & 0 & 0 \\

\vdots & \vdots & \vdots & \dots & \dots & \ddots & \vdots \\

0 & 0 & 0 &\dots & 1 & 1 & 0\\

0 & 0 & 0 &\dots & 0 & 1 & 1

\end{bmatrix}\]

The entries of $A$ is $1$ at the diagonal entries, entries below the diagonal, and $(1, n)$-entry.

The other entries are zero.

\[B=\begin{bmatrix}

1 & 0 & 0 & \dots & 0 & 0 & -1 \\

-1 & 1 & 0 & \dots & 0 & 0 & 0 \\

0 & -1 & 1 & \dots & 0 & 0 & 0 \\

\vdots & \vdots & \vdots & \dots & \dots & \ddots & \vdots \\

0 & 0 & 0 &\dots & -1 & 1 & 0\\

0 & 0 & 0 &\dots & 0 & -1 & 1

\end{bmatrix}.\]

The entries of $B$ is $1$ at the diagonal entries.

The entries below the diagonal and $(1,n)$-entry are $-1$.

The other entries are zero.

Suppose that a real matrix $A$ maps each of the following vectors

\[\mathbf{x}_1=\begin{bmatrix}

1 \\

1 \\

1

\end{bmatrix}, \mathbf{x}_2=\begin{bmatrix}

0 \\

1 \\

1

\end{bmatrix}, \mathbf{x}_3=\begin{bmatrix}

0 \\

0 \\

1

\end{bmatrix} \]
into the vectors

\[\mathbf{y}_1=\begin{bmatrix}

1 \\

2 \\

0

\end{bmatrix}, \mathbf{y}_2=\begin{bmatrix}

-1 \\

0 \\

3

\end{bmatrix}, \mathbf{y}_3=\begin{bmatrix}

3 \\

1 \\

1

\end{bmatrix},\]
respectively.

That is, $A\mathbf{x}_i=\mathbf{y}_i$ for $i=1,2,3$.

Find the matrix $A$.

(*Kyoto University Exam*)

Read solution

Let $a$ and $b$ be two distinct positive real numbers. Define matrices

\[A:=\begin{bmatrix}

0 & a\\

a & 0

\end{bmatrix}, \,\,

B:=\begin{bmatrix}

0 & b\\

b& 0

\end{bmatrix}.\]

Find all the pairs $(\lambda, X)$, where $\lambda$ is a real number and $X$ is a non-zero real matrix satisfying the relation

\[AX+XB=\lambda X. \tag{*} \]

(*The University of Tokyo Linear Algebra Exam*)

Let $A$ be a $4\times 4$ real symmetric matrix. Suppose that $\mathbf{v}_1=\begin{bmatrix}

-1 \\

2 \\

0 \\

-1

\end{bmatrix}$ is an eigenvector corresponding to the eigenvalue $1$ of $A$.

Suppose that the eigenspace for the eigenvalue $2$ is $3$-dimensional.

**(a)** Find an orthonormal basis for the eigenspace of the eigenvalue $2$ of $A$.

**(b)** Find $A\mathbf{v}$, where

\[ \mathbf{v}=\begin{bmatrix}

1 \\

0 \\

0 \\

0

\end{bmatrix}.\]

(*The University of Tokyo Linear Algebra Exam*)

Find $A^{10}$, where $A=\begin{bmatrix}

4 & 3 & 0 & 0 \\

3 &-4 & 0 & 0 \\

0 & 0 & 1 & 1 \\

0 & 0 & 1 & 1

\end{bmatrix}$.

(*Harvard University Exam*)

Find a basis for the subspace $W$ of all vectors in $\R^4$ which are perpendicular to the columns of the matrix

\[A=\begin{bmatrix}

11 & 12 & 13 & 14 \\

21 &22 & 23 & 24 \\

31 & 32 & 33 & 34 \\

41 & 42 & 43 & 44

\end{bmatrix}.\]

(*Harvard University Exam*)

Suppose that $A$ is a diagonalizable matrix with characteristic polynomial

\[f_A(\lambda)=\lambda^2(\lambda-3)(\lambda+2)^3(\lambda-4)^3.\]

**(a)** Find the size of the matrix $A$.

**(b)** Find the dimension of $E_4$, the eigenspace corresponding to the eigenvalue $\lambda=4$.

**(c)** Find the dimension of the kernel(nullspace) of $A$.

(*Stanford University Linear Algebra Exam*)

Let $A$ be an $m \times n$ real matrix.

Then the* kernel* of $A$ is defined as $\ker(A)=\{ x\in \R^n \mid Ax=0 \}$.

The kernel is also called the* null space* of $A$.

Suppose that $A$ is an $m \times n$ real matrix such that $\ker(A)=0$. Prove that $A^{\trans}A$ is invertible.

(*Stanford University Linear Algebra Exam*)

Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues.

Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix.

(*Stanford University Linear Algebra Exam*)

See below for a generalized problem.

Add to solve laterIf $L:\R^2 \to \R^3$ is a linear transformation such that

\begin{align*}

L\left( \begin{bmatrix}

1 \\

0

\end{bmatrix}\right)

=\begin{bmatrix}

1 \\

1 \\

2

\end{bmatrix}, \,\,\,\,

L\left( \begin{bmatrix}

1 \\

1

\end{bmatrix}\right)

=\begin{bmatrix}

2 \\

3 \\

2

\end{bmatrix}.

\end{align*}

then

**(a)** find $L\left( \begin{bmatrix}

1 \\

2

\end{bmatrix}\right)$, and

**(b)** find the formula for $L\left( \begin{bmatrix}

x \\

y

\end{bmatrix}\right)$.

If you think you can solve (b), then skip (a) and solve (b) first and use the result of (b) to answer (a).

(Part (a) is an exam problem of *Purdue University*)

Let $A$ be an $n$ by $n$ matrix with entries in complex numbers $\C$. Its only eigenvalues are $1,2,3,4,5$, possibly with multiplicities. What is the rank of the matrix $A+I_n$, where $I_n$ is the identity $n$ by $n$ matrix.

(*UCB-University of California, Berkeley, Exam*)