For $2 \times 2$ matrix, if $AB = -BA$, what properties would $A$ and $B$ have with regards to each other?

Asked Jan 27 '14 at 12:23

Active Jan 27 '14 at 12:23

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As title says, assume that $A$ and $B$ are two-by-two matrices. I want them to satisfy $AB = -BA$. What properties do these two matrices have with regards to each other? Sufficient properties would be fine; it doesn't have to be necessary conditions.

Also, $AB \neq 0$.

asked Jan 27 '14 at 12:23

Linear

For necessay conditions see http://math.stackexchange.com/questions/185457/matrices-conditions-for-abba-0 – user127.0.0.1 Jan 27 '14 at 12:33
@user127.0.0.1: Like $A=B=I_2$??? – Marc van Leeuwen Jan 27 '14 at 13:03
@MarcvanLeeuwen of course, thanks. for idempotent matrices it holds $AB-BA=0$ not $AB+BA=0$ – user127.0.0.1 Jan 27 '14 at 13:07
Note that we get $ABA = A(BA) = A(-AB)= -A^2 B$ so if $b$ is invertible then $B A B^{-1} = -A$: $A$ is similar to $-A$, so all eigenvalues of $A$ must be zero. So only possible interesting solutions are singular. – kjetil b halvorsen Jan 27 '14 at 13:52

For $2 \times 2$ matrix, if $AB = -BA$, what properties would $A$ and $B$ have with regards to each other?

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