# 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 $A\in \mathbb{R}^n$.

1. Since the multiplication of all eigenvalues is equal to the determinant of the matrix, $A$ full rank is equivalent to $A$ nonsingular.
2. The above also implies $A$ has linearly independent rows and columns. So $A$ is invertible.
3. $A$ is diagonalizable iff $A$ has $n$ linearly independent eigenvectors. ($A$ is nondefective).
Note: $A$ is defective if geo. multiplicity $<$ alge. multiplicity.
4. 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?

$1.$

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-

1. Characteristic polynomial splits or not,
2. $n-rank(M-\lambda_iI)=$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?

• What do you mean 'splits'? And in your answer, how do you link between diagonalizable and nonsingular(full rank) after you say no? Or simply speaking, is there any example to say $A$ is full rank but not diagonalizable? – sleeve chen Jun 4 '16 at 9:30
• If $A$ is full rank, then there is no eigenvalues $=0$. Therefore, we have only to consider $U$ if $A = U^{-1}\Lambda U$ Then can we say $U$ is full rank by knowing that $A$ is full rank? – sleeve chen Jun 4 '16 at 9:33
• A polynomial splits if it can be written as a product of linear polynomials. This depends on the field you're working in, e.g. $X^2 - 2$ splits in $\mathbb{R}$, but not in $\mathbb{Q}$. – Bib-lost Jun 4 '16 at 9:35
• $\begin{bmatrix} 1& 1 \\ 0& 1 \end{bmatrix}$ is Full rank, but not diagonalizable. – Bib-lost Jun 4 '16 at 9:38

Full rank and diagonalizability are independent from one another.

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

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

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

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