General form for $A^2=0$ where $A$ is a $2\times2$ matrix

Question

Find the general form for all the $2\times2$ matrices over $\mathbb{R}$ that $A^2=0$

Answer 1

suppose $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, then $A^2=0$ makes a four equation group of four unknown numbers($a,b,c,d$) like: $$a^2+bc = 0\\ab+bd = 0\\ac+cd = 0\\bc+d^2 = 0$$

solve these equations, we have two possible solutions: $$a=0, b=0,c\in R, d=0$$ or $$a\in R,b\in R,c = -a^2/b,d=-a.$$

So the forms of matrix are: $$\begin{bmatrix}a&b\\-a^2/b&-a\end{bmatrix}$$ or $$\begin{bmatrix}0&0\\c&0\end{bmatrix}$$

Answer 2

Either $A=0$, or $A\neq 0$ and there exists $x\in\mathbb R^2$ such that $\{x,Ax\}$ is a basis of $\mathbb R^2$. We thus have the general form of $A$ up to a change of basis.

Answer 3

Hint: First show that both eigenvalues of $A$ are $0$. Then, find the Jordan form of $A$. There are $2$ possibilities.

Answer 4

The characteristic polynomial of the given matrix is X^2=0 So obviously, the eigenvalues can only be zeros. So it's Jordan canonical form J is either [0,0;0,0] or [0,1;0,0]. Further, the matrix M=P\J P , for any non singular 2by 2 matrix P.

Answer 5

The four condition that Martial found boil down to $\begin{pmatrix}\sqrt{-bc}& b\\ c & -\sqrt{-bc}\end{pmatrix}$ where $bc\le0$.