Common solution of $Ax=0$ and $Bx=0$ Could anyone tell me When $Ax=0=Bx$ has a common non trivial solution when $A,B\in\mathbb{R}^{m\times n}$
Suppose $x_1\ne 0$ be that common solution, then what we get is $Ax_1=Bx_1=0$ but I am not getting any relation between them!
are they similar? does there exists non singular matrix $P$ such that $P^{-1}AP=B$?

They don't have to be similar. Let $A=\pmatrix{1&-1\cr1&-1\cr}$, $B=\pmatrix{1&-1\cr2&-2\cr}$. Then $x=(1,1)$ is in the nullspace of both matrices, but the two matrices are not similar, as they don't have the same eigenvalues. 
