question about eigenvectors and eigenvalues question (linear algebra)?

2012-07-19 4:34 pm

A projection is a linear operator P such that P2 = P. Let v be an
eigenvector with eigenvalue λ for a projection P, what are all possible
values of λ? Show that every projection P has at least one eigenvector.
Note that every complex matrix has at least 1 eigenvector, but you
need to prove the above for any field

Thanks!

回答 (1)

[匿名]

2012-07-19 5:43 pm

✔ 最佳答案

For a projection P, λ = 1 or 0 for every eigenvector.

Let P be a projection operator, and let v be a non-zero matrix.
Suppose that P(v) = 0, then v is an eigenvector with λ = 0.
Suppose, alternatively, that P(v) = w ≠ 0. Then:
P(w) =
P(P(v)) =
P^2(v) =
P(v) =
w

So, w is an eigenvector with λ=1.

QED

👍 0 👎 0

收錄日期: 2021-04-28 14:13:45
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20120719083401AA1dXa6

檢視 Wayback Machine 備份