Suppose that $A$ is an $m \times n$ matrix with $m > n$. Now consider a matrix $B$ which is formed by adding new rows to the matrix $A$. Is it possible for the column rank of $B$ to be smaller than that of $A$?
I cannot see why this is possible. I would expect that by increasing the number of rows, the column rank should either increase or stay the same. That is, you're either adding linearly independent or dependent rows with the latter not changing the rank.
I am asking because I've tried this numerically in Matlab and I saw that the rank can decrease if rows are added. I am not sure if this is just a numerical issue especially considering that I'm taking the rank of a 400000 by 10 matrix…
Best Answer
No. Note that adding rows can only increase the row rank. Now use the theorem that row rank equals column rank.
I don’t have a good explanation for why Matlab is acting strangely, but I strongly recommend that you check some small examples before slamming your CPU with the scary large computations.