Eigenvectors of a Homothety operator

Question

Prove that every non-zero vector is an eigenvector of the Linear Operator L

$\iff$

L is the Homothety Operator v $\rightarrow \alpha v\;$ where $\alpha$ is some fixed scalar

One direction is trivial but im having a hard time proving that every non zero vector is an eigenvectors implies that it's a homothety.

Help Please

score 2 · Accepted Answer · answered Mar 08 '15 at 21:20

2

Suggestion: It's probably easier to approach the "hard" direction as a contrapositive. So, suppose $v$ and $w$ are eigenvectors of $L$ with distinct eigenvalues. What can you say about $v + w$? (Particularly, is it an eigenvector of $L$?)

answered Mar 08 '15 at 21:20

Andrew D. Hwang

78,195

score 1 · Answer 2 · answered Mar 08 '15 at 21:42

1

Hint: What is the matrix of the operator in an arbitrary basis?

answered Mar 08 '15 at 21:42

Neal

32,659

Eigenvectors of a Homothety operator

2 Answers2

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