Whether a low-rank matrix is still low-rank after deleting one row?

Question

There is a $n\times p$ matrix $A$ with rank $r(<<n,p)$. If we abort any one row of it but keep the left rows as a new matrix, for example, $A_{-i}$ (here $i$ identifies the row number aborted), is $A_{-i}$ still low-rank? If not, how low-rank is it?

Answer 1

The matrix is at most rank $r$ and at worst rank $r-1$.

The rank of a matrix is the number of linearly independent columns it has. So, if you remove one column, then you have two options:

• If the columns was linearly independent of the rest, then the number of linearly independent columns has decreased by one. Therefore, the new rank is $r-1$.
• If the columns was a linear combination of some other columns, then the number of linearly independent columns remained the same. Therefore, the rank of the new matrix is $1$.

For example, if I have the $1\times 2$ matrix $\begin{bmatrix}1 & 0\end{bmatrix}$ (which is of rank $1$) and I remove one column, then the rank of the new matrix will either be $1$ (if I remove the second column) or $0$ (if I remove the first column).