I have been struggling with proving a relation regarding the smallest number of linearly dependent columns of product of matrices. Any insights would be greatly appreciated.
Let $m$ and $n$ be the smallest number of linearly dependent columns of matrices $A$ and $B$ respectively. Then, what can we say about the smallest number of linearly dependent columns of the product matrix $AB$?
An analogous property exists for the largest number of columns that are linearly independent; in other words, the rank of the matrix i.e. $rank(AB) \leq min(rank(A), rank(B))$.
Can we say something similar about the smallest number of linearly dependent columns of the product of two matrices?
Edit:
As pointed out by user164568, this is equivalent to asking how $spark(AB)$ is related to $spark(A)$ and $spark(B)$?
-RD
Just a comment to your question: what you are referring to here is called spark of the matrix. Spark is the smallest number of linearly dependent columns of the matrix.