theory for solving eigenvector and eigenvalue that are unequal

The question asks can a vector X be an eigenvector for two unequal eigenvalues E1 and E2 for the same matrix A? If yes prove it with an example. If No explain why?


I'm really confused on this and would greatly take any and all input. Thank you.

if x is an eigenvector for the matrix A, then Ax = λ₁x, for some scalar λ₁.

if x also satisfies Ax = λ₂x, then we have: 0 = Ax - Ax = λ₁x - λ₂x = (λ₁-λ₂)x.

now eigenvectors cannot be 0 (by definition), so (λ₁-λ₂)x = 0 implies λ₁-λ₂ = 0: that is, λ₁ = λ₂.

so an eigenvector can only have ONE eigenvalue it belongs to.

it is possible, however, for an eigenvalue to have TWO (or more, linearly independent) eigenvectors.

consider the matrix I =

[1 0]
[0 1].

it should be clear that both (1,0) and (0,1) are eigenvectors with the same eigenvalue, 1, and that these two vectors are linearly independent.

Thank you Deveno! that makes sense now. It gets pretty confusing when you're looking at it for a few hours.

My next question is actually along the same lines. The question states: Can a non-zero vector X be an eigenvector for two unequal eigenvalues E1 and E2 corresponding to two different matrices A and B, respectively? If yes prove it with an example. If No explain why?

This is pretty close to trivial. Let A and B be 2 by 2 diagonal matrices with different numbers along the diagonal.

The question I stated above question where it states: Can a non-zero vector X be an eigenvector for two unequal eigenvalues E1 and E2 corresponding to two different matrices A and B, respectively? If yes prove it with an example. If No explain why?

I've setup a few matrices and I keep getting a result of zero. Can anyone give me a better example of this?

Still not getting the proper answer to the question: Can a non-zero vector X be an eigenvector for two unequal eigenvalues E1 and E2 corresponding to two different matrices A and B, respectively? If yes prove it with an example. If No explain why? Can someone please help! Thank you again in advance for your time.