Can a Complex Matrix X Solve the Column Subspace Problem?

Then we can fix that by filling in the rest of the rows to make the matrix square.In summary, the question is how to find a complex matrix X with size M by N such that the span of A_i*X is equal for all i from 1 to k. The conditions for the existence of nontrivial solutions are related to the sizes of M, N, and k. There are methods and references available for this question.
  • #1
wangxianfeng
1
0
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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  • #2
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.
 

What is the column subspace problem?

The column subspace problem is a mathematical problem that involves finding a set of vectors that span a given subspace. In other words, it is the process of determining all possible linear combinations of a set of vectors that can be used to create other vectors within a given subspace.

Why is the column subspace problem important in science?

The column subspace problem is important in science because it is used in various fields, such as physics, engineering, and computer science, to model and solve complex systems. It allows scientists to represent a large number of variables in a more compact and efficient way, making it easier to analyze and make predictions.

What are the applications of the column subspace problem?

The column subspace problem has many applications, including image and signal processing, data compression, and machine learning. It is also used in solving systems of linear equations, which is a common problem in many scientific fields.

What are the challenges of solving the column subspace problem?

One of the main challenges of solving the column subspace problem is that it can be computationally expensive, especially for large datasets. Another challenge is that the solution may not always be unique, leading to the need for additional constraints or methods to determine the most suitable solution.

How is the column subspace problem solved?

The column subspace problem is typically solved using algorithms, such as the Gram-Schmidt process or the QR decomposition. These algorithms use mathematical techniques to find a set of orthogonal basis vectors that span the subspace, making it easier to solve for the desired linear combinations.

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