Linearly dependent eigenvectors of a matrix I read a theorem that says squared matrix $A_{n\times n}$ is diagonalizable iff there is a set of $n$ linearly independent vectors ,each of which is an eigen-vector of $A_{n\times n}$ .
I understand from the theorem that if $A_{n\times n}$ has at least two dependent eigen-vectors
then $A_{n\times n}$ is not diagonalizable. Is it true?and according to this I want to understand when a matrix has linearly dependent eigenvectors ? (eigenvectors is a set of vectors)
In other words if there is'nt any set of linearly independent set of eigenvectors for a matrix, then does that mean that the matrix is not diagonalizable? 
 A: The theorem states, correctly, that IF the matrix $A$ has $n$ linearly independent eigenvectors, then $A$ is diagonalizable.
That does NOT mean that if $A$ has two dependent eigenvectors, that it is not diagonalizable. In fact, if $A$ has at least one eigenvector $x$ such that $Ax=\lambda x$, then $2x$ is also an eigenvector of $A$ since $A(2x)=2Ax=2\lambda x=\lambda (2x)$. And since $\{x,2x\}$ is not a linearly independent set, this means that $A$ has two linearly dependent vectors. Does that mean $A$ is not diagonalizable? Of course not!
You actually made two logical mistakes.
You know that if statement $p$ ($A$ has $n$ linearly dependent eigenvectors) is true, then statement $q$ ($A$ is diagonalizable) is true, so in other words $p\implies q$. Then you found another statement, $c$ ($A$ has two linearly independent eigenvectors).
Then, you made two mistakes:


*

*You claim that if $c$ is true, then $p$ is not true (in fact, any matrix with at least one eigenvector has two dependent eigenvectors, but it can still have $n$ linearly dependent eigenvectors. In fact, whenever $p$ is true, $c$ is also true!

*You then wanted to conclude that since $p$ is not true, then $q$ is not true. This is a very common mistake, but if $p\implies q$, then $\neg p$ does not imply $\neg q$! For example, the statement "Rex is a dog" being true means that the statement "Rex has fur" is true, however, the statement "Rex is not a dog" does not mean that "Rex has no fur" is true!

