Jordan canonical form of a matrix $A$ with the property that $A^2 = 0$?

Question

Suppose $A \in \mathbb{C}^{2 \times 2}$ has the property that $A^2 = 0$. What are the possible Jordan canonical forms of such a matrix?

A matrix of the form:

\begin{pmatrix} a & a \\ -a & -a \\ \end{pmatrix}

for $a \in \mathbb{C}$ will satisfy the $A^2 = 0$ property. The characteristic polynomial of this matrix will be $(t-a)(t+a) + a^2 = t^2$ So the eigenvalues will be $0$. Are there other possibilities?

Answer 1

$A^2=0$, so $A$ is nilpotent, so its only possible eigenvalue is $0$. So there are only $2$ possible Jordan canonical form : $$\begin{pmatrix} 0 & 0 \\ 0 & 0 \\ \end{pmatrix}\text{ and }\begin{pmatrix} 0 & 1 \\ 0 & 0 \\ \end{pmatrix}$$

Answer 2

In dimension $2$ one has the simplest case for the matrix in Jordan form will be (assuming it is not the zero matrix)

$$\begin{bmatrix} 0 & 1\\0 & 0 \end{bmatrix}$$