# find-eigenvalues-eigenvectors

by Yu · Published · Updated

Add to solve later

Add to solve later

Add to solve later

### More from my site

- Express a Vector as a Linear Combination of Given Three Vectors Let \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ 4 \\ 3 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \mathbf{b}=\begin{bmatrix} 2 \\ 13 \\ 6 […]
- In a Principal Ideal Domain (PID), a Prime Ideal is a Maximal Ideal Let $R$ be a principal ideal domain (PID) and let $P$ be a nonzero prime ideal in $R$. Show that $P$ is a maximal ideal in $R$. Definition A commutative ring $R$ is a principal ideal domain (PID) if $R$ is a domain and any ideal $I$ is generated by a single element […]
- Find a Basis and Determine the Dimension of a Subspace of All Polynomials of Degree $n$ or Less Let $P_n(\R)$ be the vector space over $\R$ consisting of all degree $n$ or less real coefficient polynomials. Let \[U=\{ p(x) \in P_n(\R) \mid p(1)=0\}\] be a subspace of $P_n(\R)$. Find a basis for $U$ and determine the dimension of $U$. Solution. […]
- Diagonalize the $2\times 2$ Hermitian Matrix by a Unitary Matrix Consider the Hermitian matrix \[A=\begin{bmatrix} 1 & i\\ -i& 1 \end{bmatrix}.\] (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, find the eigenvectors. (c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix […]
- A Subgroup of the Smallest Prime Divisor Index of a Group is Normal Let $G$ be a finite group of order $n$ and suppose that $p$ is the smallest prime number dividing $n$. Then prove that any subgroup of index $p$ is a normal subgroup of $G$. Hint. Consider the action of the group $G$ on the left cosets $G/H$ by left […]
- Eigenvalues of a Stochastic Matrix is Always Less than or Equal to 1 Let $A=(a_{ij})$ be an $n \times n$ matrix. We say that $A=(a_{ij})$ is a right stochastic matrix if each entry $a_{ij}$ is nonnegative and the sum of the entries of each row is $1$. That is, we have \[a_{ij}\geq 0 \quad \text{ and } \quad a_{i1}+a_{i2}+\cdots+a_{in}=1\] for $1 […]
- Multiplicative Groups of Real Numbers and Complex Numbers are not Isomorphic Let $\R^{\times}=\R\setminus \{0\}$ be the multiplicative group of real numbers. Let $\C^{\times}=\C\setminus \{0\}$ be the multiplicative group of complex numbers. Then show that $\R^{\times}$ and $\C^{\times}$ are not isomorphic as groups. Recall. Let $G$ and $K$ […]
- Every Cyclic Group is Abelian Prove that every cyclic group is abelian. Proof. Let $G$ be a cyclic group with a generator $g\in G$. Namely, we have $G=\langle g \rangle$ (every element in $G$ is some power of $g$.) Let $a$ and $b$ be arbitrary elements in $G$. Then there exists […]