Can a Complex Matrix X Solve the Column Subspace Problem?

[wangxianfeng]

Let A_i (i=1,...,k) be a nonsingular complex matrix which size is M by M.
The question is how to find a complex matrix X which size is M by N such that:

span(A_1*X)=...=span(A_k*X)

(I guess that there must be relations between M,N and k when nontrival solution exists. )
ask:
1)if non trival solwhat's the condition for the existence of nontrival solutions?
2)Is there any method or related references for this question?

Thanks in advance:)

 

[HallsofIvy]

It seems to me that a rather obvious thing to do is to add N-M rows that are just multiples, or linear combinations, of the original rows.