Eigen Value & Eigen Vector Pairwise Relationship

Question

Having same eigen values implies eigen vectors are linearly dependent. But why does it not imply that the eigen vectors are same? Are the eigen value and eigen vector pairs not unique for non-zero eigen values?

Answer 1

The Eigenspace associated with an eigenvalue is unique, but of course there are infinitely many eigenvectors in such a space and no basis (for a vector space/subspace) is unique. I am not exactly sure what is meant by "having same eigen values" - do you mean an eigenvalue with algebraic multiplicity greater than 1 for a certain operator? If so, note that it is possible to have linearly independent eigenvectors associated with such an eigenvalue...the eigenspace dimension is less than or equal to the algebraic multiplicity of the eigenvalue.

Answer 2

The first statement is not correct. It is possible for two linearly independent vectors to be eigenvectors of the same eigenvalue. For an extreme example, try the identity or the zero transformation.

What is true is that eigenvectors of different eigenvalues are linearly independent.