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:)
 
Physics news on Phys.org
  • #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.
 

FAQ: Can a Complex Matrix X Solve the Column Subspace Problem?

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.

Similar threads

MHB Is $s$ the unique vector that spans the solution space $L(A,0)$?
Replies
3
Views
1K
I How to Construct an Orthonormal Basis for a 2D Subspace in Linear Algebra?
Replies
3
Views
1K
I Linear least squares regression for model matrix identification
Replies
5
Views
2K
MHB How Does the Determinant of a Matrix Relate to the Area of a Parallelogram?
Replies
15
Views
2K
I Proof that if T is Hermitian, eigenvectors form an orthonormal basis
Replies
3
Views
3K
MHB Solving Linear Equations: $Ax=b$ and Rank of A
Replies
9
Views
2K
Relationship between column space of a matrix and rref of matrix
Replies
7
Views
10K
A Eigenvalues of block matrix/Related non-linear eigenvalue problem
Replies
1
Views
1K
MHB Proving or disproving this matrix V is invertible.
Replies
1
Views
980
MHB Row reduced echelon form and its meaning
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top