If $AB+BA=0$, then $A^2B^3=B^2A^3$?

Question

If I have a matrix $A$ and $B$ such that $AB+BA=0$

is it true that $A^2B^3=B^2A^3$?

I think that it is false.

Related: http://math.stackexchange.com/questions/106495/form-of-anticommuting-matrices — vadim123, Oct 10 '14 at 03:05

Start wearing purple · Answer 1 · 2014-10-10T11:36:47.493

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It is false. Take $$A=\left(\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right),\qquad B=\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right),$$ then we have $AB+BA=0$, but also $A^2=B^2=1$ so that $A^2B^3=B$ and $B^2A^3=A$.

edited Oct 10 '14 at 11:36

answered Oct 10 '14 at 03:05

Start wearing purple

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Neat counter-example. +1 – Cameron Williams Oct 10 '14 at 03:05
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How did you find this? – Dan Uznanski Oct 10 '14 at 03:07
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@DanUznanski For a physicist, the most natural example of anticommuting matrices is given by Pauli matrices. Otherwise one could of course just multiply $A$ by a scalar. – Start wearing purple Oct 10 '14 at 03:10
Another way now that I've seen @ChrisCutler's answer is that these two matrices are equivalent to the quaternions $k$ and $i$ respectively, using the 2x2 matrix in $\mathbb C$ form. Or that times -i, in any case. – Dan Uznanski Oct 10 '14 at 03:12
@O.L. I should keep the Pauli matrices in mind. I'm ashamed of myself as a former physicist. I was having trouble coming up with examples of commuting matrices... – Cameron Williams Oct 10 '14 at 03:20

score 4 · Answer 2 · answered Oct 10 '14 at 03:08

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Similarly to O.L.'s answer, we have the quaternions $ij+ji=k-k=0$, but $i^2j^3=j$ and $j^2i^3=i$.

answered Oct 10 '14 at 03:08

Chris Culter

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If $AB+BA=0$, then $A^2B^3=B^2A^3$?

2 Answers2