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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

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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.

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  • $\begingroup$ Thanks. I have modified the question to reflect this. $\endgroup$
    – r2d2
    Jan 1, 2016 at 20:25

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