# find-eigenvalues-eigenvectors

by Yu · Published · Updated

Add to solve later

Add to solve later

Add to solve later

### More from my site

- The Column Vectors of Every $3\times 5$ Matrix Are Linearly Dependent (a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly dependent. (b) Prove that the row vectors of every $5\times 3$ matrix $B$ are linearly dependent. Proof. (a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly […]
- Determinant of a General Circulant Matrix Let \[A=\begin{bmatrix} a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\ a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\ a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\ \vdots & \vdots & \dots & \vdots & \vdots \\ a_{2} & a_3 & \dots & a_{0} & a_{1}\\ a_{1} & a_2 & […]
- Quiz 6. Determine Vectors in Null Space, Range / Find a Basis of Null Space (a) Let $A=\begin{bmatrix} 1 & 2 & 1 \\ 3 &6 &4 \end{bmatrix}$ and let \[\mathbf{a}=\begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix} 1 \\ 1 […]
- The Determinant of a Skew-Symmetric Matrix is Zero Prove that the determinant of an $n\times n$ skew-symmetric matrix is zero if $n$ is odd. Definition (Skew-Symmetric) A matrix $A$ is called skew-symmetric if $A^{\trans}=-A$. Here $A^{\trans}$ is the transpose of $A$. Proof. Properties of […]
- Determine Trigonometric Functions with Given Conditions (a) Find a function \[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)\] such that $g(0) = g(\pi/2) = g(\pi) = 0$, where $a, b, c$ are constants. (b) Find real numbers $a, b, c$ such that the function \[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 […]
- If Squares of Elements in a Group Lie in a Subgroup, then It is a Normal Subgroup Let $H$ be a subgroup of a group $G$. Suppose that for each element $x\in G$, we have $x^2\in H$. Then prove that $H$ is a normal subgroup of $G$. (Purdue University, Abstract Algebra Qualifying Exam) Proof. To show that $H$ is a normal subgroup of […]
- Possibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a) \[\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right. \] where $a,b,c, d$ […]
- Example of an Infinite Group Whose Elements Have Finite Orders Is it possible that each element of an infinite group has a finite order? If so, give an example. Otherwise, prove the non-existence of such a group. Solution. We give an example of a group of infinite order each of whose elements has a finite order. Consider […]