特征值与特征向量

Eigenvalues and Eigenvectors

Introduction

Eigenvalues and Eigenvectors

Eigenspaces

The Invertible Matrix Theorem: Addenda

Let $A = \begin{pmatrix} 2 & 2 \\ -4 & 8 \end{pmatrix}$. Which of the following is an eigenvector of $A$?

Let $A = \begin{pmatrix} 1 & 3 \\ 2 & 6 \end{pmatrix}$. Which statement is true?

If $Av = \lambda v$ for some nonzero vector $v$, then the scalar $\lambda$ is called an ______ of $A$.

Let $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ (reflection across the line $y=x$) and $v = \begin{pmatrix} 3 \\ -3 \end{pmatrix}$. Compute $Av$ and enter the eigenvalue of $A$ for this eigenvector (a number).

A $3\times3$ matrix $B$ is known to have eigenvalues $0$, $1$ and $2$. Enter the rank of $B$ (a number).

Let $\lambda$ be an eigenvalue of a square matrix $A$. The eigenspace of $A$ corresponding to $\lambda$ is defined as which set?

Let $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$. The number $\lambda = 2$ is an eigenvalue. Enter a basis vector of the eigenspace as $x,y$.