A matrix acts on each vector below as noted. Sort: which are eigenvectors?

Groups: Eigenvector · Not an eigenvector

$Av = v + (1,0)$ (shifted sideways)
$Av = 0$ (annihilated), $v \ne 0$
$Av = -v$ (exactly reversed)
$Av = 5v$ (stretched in place)

Hints

The test is strict proportionality: $Av = \lambda v$ for some number $\lambda$ , which may be negative or even zero.
Reversal is scaling by $-1$ ; annihilation is scaling by $0$ . Shifting, however, is not scaling.

Show the answer

Eigenvector: $Av = 5v$ (stretched in place), $Av = -v$ (exactly reversed), $Av = 0$ (annihilated), $v \ne 0$

Not an eigenvector: $Av = v + (1,0)$ (shifted sideways)

Why

Eigenvalue $\lambda$ can be any scalar: $-1$ flips, $0$ crushes (such $v$ spans the null space). The only disqualifier is leaving the original line, proportionality is the entire definition.

Read the lesson: Eigenvalues and Eigenvectors →

Practise Eigenvalues and Eigenvectors

The app has 8 more questions on this lesson, and keeps your place in the course. Physics I is free to start.

Start Physics I free Get it onGoogle Play

A matrix acts on each vector below as noted. Sort: which are eigenvectors?

Practise Eigenvalues and Eigenvectors

More questions on Eigenvalues and Eigenvectors