Can a Complex Matrix X Solve the Column Subspace Problem?

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The discussion centers on the problem of finding a complex matrix X of size M by N that satisfies the condition span(A_1*X)=...=span(A_k*X) for nonsingular complex matrices A_i of size M by M. Key questions include the conditions for the existence of nontrivial solutions and methods or references related to this problem. It is suggested that adding N-M rows as linear combinations of the original rows may provide a solution. The relationship between the dimensions M, N, and the number of matrices k is also highlighted as critical for determining solution existence.

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wangxianfeng
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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:)
 
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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.
 

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