Diagonalizing Linear Operators: Understanding the Differences

[db1uover]

Homework Statement

Let V be a n-dimensional real vector space and L: V --> V be a linear operator. Then,
A.) L can always be diagonalized
B.) L can be diagonalized only if L has n distinct eigenvalues
C.) L can be diagonalized if all the n eigenvalues of L are real
D.) Knowing the eigenvalues is always enough to decide if L can be diagonalized or not
E.) L can be diagonalized if all its n eigenvalues are distinct

Homework Equations

I have the answer, but I don't understand the slight differences between 'only if L has n distinct eigenv' and 'if all its n eigenv are distinct'. Can someone explain how the statements have different meanings?

The Attempt at a Solution

Starting with the assumption that the statements are different, I understood it as Ans B means must have only n eigenv while Ans E is slightly more flexible in its requirements. Is this correct? Is there more to the story?

 

[vela]

B means L can be diagonalized only if all of its eigenvalues are distinct; if any are repeated, you can't diagonalize L.

E means if the eigenvalues are distinct, then you can diagonalize L. It doesn't say anything about the case of repeated eigenvalues.