[Math] Is a matrix diagonalizable, if one of its eigenvalues is zero

Question

I checked weather the following matrix is diagonalizable.

$$A=\begin{bmatrix}
4 & 0 & 4\\
0 & 4 & 4\\
4 & 4 & 8
\end{bmatrix}$$

And the corresponding eigenvalues were $0$, $4$, $12$.

Now if we write the similar diagonal matrix to $A$ it would be,

$$D=\begin{bmatrix}
0 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & 12
\end{bmatrix}$$

which is theoretically not a diagonal matrix.

Now is $A$ diagonalizable?

Answer 1

$D$ is a diagonal matrix. A square matrix is a diagonal matrix if and only if the off-diagonal entries are $0$.

Hence your matrix is diagonalizable.

In fact, if the eigenvalues are all distinct, then it is diagonalizable.