Thread: eigen vector problem....

Q:Let u and v be the eigen vectors of A corresponding to eigen values 1 and 3 respectively.Prove that u+v is not an eigen vector of A?

Think

Originally Posted by Gamma

Think

really very silly problem...

Originally Posted by Mathventure

Q:Let u and v be the eigen vectors of A corresponding to eigen values 1 and 3 respectively.Prove that u+v is not an eigen vector of A?

Since (1,u) and (3,v) are eigenpairs, then and , where

Suppose that is an eigenvector. Thus where

But and this is equal to . Comparing both sides, we get , which is impossible!

It is now evident that has no eigenvalue, and thus it is not an eigenvector of A.

Originally Posted by Mathventure

Q:Let u and v be the eigen vectors of A corresponding to eigen values 1 and 3 respectively.Prove that u+v is not an eigen vector of A?

let's do the general case: suppose where are non-zero vectors and are scalars. we'll show that there's no scalar satisfying

so suppose there's such then and thus call this 1). we also have which with 1) gives us:

and hence because and this with 1) gives us which is a contradiction!


Note: the above is a trivial thing if you know that distinct eigenvalues correspond to linearly independent eigenvectors.