Given $B^T=B$ and if
$A^{T}BA=0,$
with $B\in{\rm{M}}_{2\times2}(\mathbb{C})$ and $A\in{\rm{M}}_{2\times2}(\mathbb{R})$
what values may $B$ take to satisfy this equation?
I think $B=0$ is one solution, any others?
more questions: just yes/ no answer is okay for these :)
if $ABC=0$ then does $(ABC)^t=0^t=0$
where $X^t$ is the transpose of $X$