# Satisfying $AB=0$, commute and $A^2+B^2=0$

I heard somewhere that $AB=0$ is related to $A^2+B^2$.

So, does $AB=0$ result in $A^2+B^2 =0$?

Or if it doesn't, which matrices would satisfy $AB=0$ while $A^2+B^2=0$

Edit: right. stupid me. So, let me add the following condition:

Suppose $AB=0, CD=0, EF=0....$ and $A^2+B^2=0, C^2+D^2=0, E^2+F^2=0....$. Except zero matrix, is there any case when a set of matrices have the aforementioned property?

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$$\pmatrix{0&1\\0&0}\pmatrix{1&0\\0&0}=\pmatrix{0&0\\0&0}\;,$$

but

$$\pmatrix{0&1\\0&0}^2+\pmatrix{1&0\\0&0}^2=\pmatrix{0&0\\0&0}+\pmatrix{1&0\\0&0}=\pmatrix{1&0\\0&0}\ne\pmatrix{0&0\\0&0}\;.$$

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Counterexample

$$A=\begin{pmatrix}1&0\\0&0\end{pmatrix}\;\;,\;\;B=\begin{pmatrix}0&0\\1&0\end{pmatrix}$$

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Hehe...yeah. Already fixed it, thanks. – DonAntonio Oct 28 '12 at 11:44

Since $AB = 0$ does not imply $A = 0$ or $B =0$, unlike real numbers, you can find such a matrices that satisfy $A^2 + B^2 = 0$. But it doesnt hold in general.

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If $$A=B=C=D=\dots=\pmatrix{0&1\cr0&0\cr}$$ then $AB=CD=EF=\dots=0$ and $A^2+B^2=C^2+D^2=E^2+F^2=\dots=0$

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One example satisfying $AB=0$ and $A^2+B^2=0$ would be $$A=\begin{bmatrix}0&0&0&0\\1&0&0&0\\c&0&0&0\\0&1&0&0\end{bmatrix} \qquad B=\begin{bmatrix}0&0&0&0\\0&0&0&0\\-1&0&0&0\\0&0&1&0\end{bmatrix}$$ You get $BA=0$ additionally iff $c=0$. All smaller examples must additionally satisfy $A^2=B^2=0$.

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