# find-eigenvalues-eigenvectors

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• All the Conjugacy Classes of the Dihedral Group $D_8$ of Order 8 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 $8$. Hint. You may directly compute the conjugates of each element but we are going to use the following theorem to simplify the […]
• Diagonalize a 2 by 2 Symmetric Matrix Diagonalize the $2\times 2$ matrix $A=\begin{bmatrix} 2 & -1\\ -1& 2 \end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   Solution. The characteristic polynomial $p(t)$ of the matrix $A$ […]
• Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent By calculating the Wronskian, determine whether the set of exponential functions $\{e^x, e^{2x}, e^{3x}\}$ is linearly independent on the interval $[-1, 1]$.   Solution. The Wronskian for the set $\{e^x, e^{2x}, e^{3x}\}$ is given […]
• A Prime Ideal in the Ring $\Z[\sqrt{10}]$ Consider the ring $\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}$ and its ideal $P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}.$ Show that $p$ is a prime ideal of the ring $\Z[\sqrt{10}]$.   Definition of a prime ideal. An ideal $P$ of a ring $R$ is […]
• Rings $2\Z$ and $3\Z$ are Not Isomorphic Prove that the rings $2\Z$ and $3\Z$ are not isomorphic.   Definition of a ring homomorphism. Let $R$ and $S$ be rings. A homomorphism is a map $f:R\to S$ satisfying $f(a+b)=f(a)+f(b)$ for all $a, b \in R$, and $f(ab)=f(a)f(b)$ for all $a, b \in R$. A […]
• The Subspace of Matrices that are Diagonalized by a Fixed Matrix Suppose that $S$ is a fixed invertible $3$ by $3$ matrix. This question is about all the matrices $A$ that are diagonalized by $S$, so that $S^{-1}AS$ is diagonal. Show that these matrices $A$ form a subspace of $3$ by $3$ matrix space. (MIT-Massachusetts Institute of Technology […]
• Subgroup Containing All $p$-Sylow Subgroups of a Group Suppose that $G$ is a finite group of order $p^an$, where $p$ is a prime number and $p$ does not divide $n$. Let $N$ be a normal subgroup of $G$ such that the index $|G: N|$ is relatively prime to $p$. Then show that $N$ contains all $p$-Sylow subgroups of […]
• Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$ Consider the complex matrix $A=\begin{bmatrix} \sqrt{2}\cos x & i \sin x & 0 \\ i \sin x &0 &-i \sin x \\ 0 & -i \sin x & -\sqrt{2} \cos x \end{bmatrix},$ where $x$ is a real number between $0$ and $2\pi$. Determine for which values of $x$ the […]