Are following statement about eigenvectors and eigenvalues correct?

Question

If there are n distinct eigenvalues, then there are eigenvectors corresponding to eigenvalues are independent.

If there are same n eigenvalues, then eigenvectors corresponding to eigenvalues are not independent.

Are above statements true?

Answer 1

Your first statement is correct. That is, the eigenvectors corresponding to different eigenvalues are linearly independent.

To address your second statement, we can discuss the geometric and algebraic multiplicities of a given eigenvalue. Given an eigenvalue $\lambda$ of a linear transformation, the algebraic multiplicity of $\lambda$ is defined to be its multiplicity as a root of the characteristic polynomial of the transformation, while the geometric multiplicity of $\lambda$ is the dimension of the $\lambda$-eigenspace. Using the above definitions, the following fact addresses your second question.

Let $\lambda$ be an eigenvalue of algebraic multiplicity $n$ and geometric multiplicity $d$. Then $1 \le d \le n$. That is, one can find at least one, but no more than $n$ linearly independent eigenvectors corresponding to the eigenvalue $\lambda$.