Diagonalizable vs full rank vs nonsingular (square matrix) There are many discussions of such type problems (comparison), for example:
Diagonalizable vs Normal 
Today, I want to clearly understand the  topic.
Suppose the matrix $A\in \mathbb{R}^n$.  


*

*Since the multiplication of all eigenvalues is equal to the determinant of the matrix, $A$ full rank is equivalent to $A$ nonsingular.    

*The above also implies $A$ has linearly independent rows and columns.  So $A$ is invertible.         

*$A$ is diagonalizable iff $A$ has $n$ linearly independent eigenvectors. ($A$ is nondefective).
Note: $A$ is defective if geo. multiplicity $<$ alge. multiplicity.    

*A diagonalizable matrix does not imply full rank (or nonsingular).


My problem is 

Does full rank matrix (nonsingular) imply it is diagonalizable?   

Equivalently:  

Does a matrix with all its columns or rows linear independently imply all its eigenvectors linear independently?      

 A: $1.$
No. A full rank matrix implies it's determinant is non-zero or the matrix is non-singular.
Speaking in terms of linear operator $T$ over a vector space $V$ is diagonalizable if and only if there exists an ordered basis consisting of eigenvectors of $T$, Furthermore if $T$ is diagonalizable and $\beta$ is an ordered basis of $T$ consisting eigen vectors of $T$, then $M=[T]_\beta$, the matrix representation of $T$, is a diagonal matrix. 
One can check if a given matrix $M_{n\times n}$ is diagonalizable or not by-


*

*Characteristic polynomial splits or not,

*$n-rank(M-\lambda_iI)=$multiplicity of $\lambda_i$.


If this two criteria fulfills then a matrix is diagonalizable.  
Counterexample: $\begin{bmatrix}3 & 1 &0 \\ 0&3&0\\ 0&0&4\end{bmatrix}$
$2$. Consider the identity matrix $I$, all it's rows and columns are linearly independent. What can you say about the eigenvectors?
A: Full rank and diagonalizability are independent from one another.


*

*A diagonalizable matrix with full rank:
$$
\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}
$$

*A non diagonalizable matrix with full rank:
$$
\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}
$$

*A diagonalizable matrix without full rank:
$$
\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}
$$

*A non diagonalizable matrix without full rank:
$$
\begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}
$$
