# Linearly dependent columns of product of matrices

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

## 1 Answer

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.

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