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Consider a standard linear regression model in a matrix form given by \begin{equation} y=X\beta+u \end{equation} where $y$ is $n \times 1$, $X$ is $n \times k$ with $n>k$. Assume that $X$ has full column rank.

When using cross validation, we typically remove some rows of $X$. Let $X_{-i}$ be an $(n-1) \times k$ matrix where $i$th row is removed from $X$. Then, even when $X$ has full column rank, $X_{-i}$ might have a reduced rank.

I'm wondering under what conditions, $X_{-i}$ is still of full column rank when $X$ is of full column rank.

Or more generally, let us consider a matrix \begin{equation} A=\begin{bmatrix} A_1 \\ A_2 \end{bmatrix} \end{equation} where $A_1$ and $A_2$ have more rows than the columns.

I'm wondering under what conditions $A_1$ becomes full rank when $A$ is of full rank.

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  • $\begingroup$ Sometimes $A_1$ is of full rank, sometimes it isn't. What kind of conditions are you looking for? We could tell you the conditions that follow from the definition of rank, but that is probably not what you are looking for. $\endgroup$
    – frank
    Oct 2, 2022 at 6:21
  • $\begingroup$ Thanks. I want to know a condition between $A_1$ and $A_2$ that ensures full rank of $A_1$. $\endgroup$
    – user0131
    Oct 2, 2022 at 6:39

1 Answer 1

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One condition could be: if all the rows of $A_2$ are in the span of the rows of $A_1$, then $A_1$ is of the same rank as $A$. But that is kind of the definition of rank.

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  • $\begingroup$ Thanks. I understand that what you write: it is true when considering the row rank. I might be missing something. But, is it true for the column rank even when rows are deleted? I'm wondering how can I prove it. $\endgroup$
    – user0131
    Oct 2, 2022 at 7:01
  • $\begingroup$ The column rank is the same as the row rank. $\endgroup$
    – frank
    Oct 2, 2022 at 7:04
  • $\begingroup$ Thanks! I was forgetting a basic result that the column rank is always equal to the row rank... $\endgroup$
    – user0131
    Oct 3, 2022 at 0:57

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