Rangesand Eigenvalues

GreenDay14 · #1 $Old$ November 8th, 2009, 06:25 PM

Hey, I have a question here that states:

a) Find T belongs to L(C²) [C being the set of complex numbers] with two distinct eigenvalues such that dim rangeT = 1.
b) Suppose T belongs to L(V) and dim rangeT = K. Prove that T has at most k+1 distinct eigenvalues.

Could anyone give me an idea on how to approach this question. Any help would be greatly appreciated. Thanks.

HallsofIvy · #2 $Old$ November 9th, 2009, 05:10 AM

Quote:

Originally Posted by GreenDay14 $View Post$

Hey, I have a question here that states:

a) Find T belongs to L(C²) [C being the set of complex numbers] with two distinct eigenvalues such that dim rangeT = 1.
b) Suppose T belongs to L(V) and dim rangeT = K. Prove that T has at most k+1 distinct eigenvalues.

Could anyone give me an idea on how to approach this question. Any help would be greatly appreciated. Thanks.

If $\lambda_1$ and $\lambda_2$ are distinct eigenvalues, then the respective eigenvectors are independent. So suppose eigenvalues $\lambda_1$ and $\lambda_2$ have eigenvectors u and v, respectively. Then u and v, separately, span different subspaces and so $Tu= \lambda_1u$ and $Tv= \lambda_2v$ span different subspaces unless one of the eigenvalues is 0!