I have read these two properties for eigen values and eigen vectors:
1) $n \times n$ Matrix A is guaranteed to have $n$ independent eigenvectors if all its eigenvalues are different.
2)If A has repeated eigenvalues, it may or may not have $n$ independent eigenvectors.
I can see how 2) might be true:
The nullspace of $(A-\lambda I )$ might have dimension $>1$, thus leading to more than one independent eigen vector $X$ when we solve $(A- \lambda I)X=0$.
Now I want to know:
If $A$ has repeated eigen values, with $m<n$ distinct eigen values, Is $A$ guaranteed to have atleast $m$ independent eigen vectors?
Equivalently, I'm trying to ask:
Is the nullspace of $(A - \lambda I)$ surely different for different values of $\lambda$?