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Consider the projection matrix in Linear Regression $P=X(X^TX)^{-1}X^T$. If we have $n$ points, $P$ is an $n$ x $n$ matrix. We also know it satisfies a number of properties, including that it's symmetric, idempotent, and if $X$ is $n$ x $r$, then $P$ has rank $r$. More here: https://en.wikipedia.org/wiki/Projection_matrix

Now consider the principle sub-matrix we get when removing any arbitrary $n-r$ rows and the corresponding columns. For example, if we remove row 1, 3, 5 we must also remove column 1, 3, 5. Call this new matrix $G$. We know $G$ is $r$ x $r$ and is symmetric.

We want to also show that $G$ has rank $r$. That is, by removing any arbitrary $n-r$ rows and the corresponding columns from $P$, the resulting principle sub-matrix has rank $r$.

I know if a matrix $A$ has rank $k$, than any sub-matrix of A must have rank $\leq k$. But that doesn't seem very helpful here for this proof. Intuitively, it seems that this statement should hold. I tried this out on Mathematica, and holds for $n=3$ and $n=4$ (with $r=2$). I'm not exactly sure how to proceed in trying to show this rigorously.

Edit: $X$ is the design matrix in regression, which always has a column of 1s at the end. So say we are in 2d, and we have the following $(x,y)$ points that we want to fit a line through: (2,1), (4,2), (5,9). Then the $X$ matrix looks like: $$ \begin{matrix} 2 & 1 \\ 4 & 1 \\ 5 & 1\\ \end{matrix} $$ In regression, we want to learn the equation of line (or hyperplane for higher dimension) which has a y-intercept term. Once we have such a $\beta$, our projection/regression is given by $X\beta$. The column of 1 is to account for the y-intercept term. Generally speaking $X$ has rank $r$.

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I don't think what you want is true, unless you have further conditions on $X$. Let $$ X=\begin{bmatrix} 1&0\\0&1\\ 0&0\end{bmatrix}. $$ Then $n=3$, $r=2$, and $$ P=X(X^TX)^{-1}X=\begin{bmatrix} 1&0\\0&1\\ 0&0\end{bmatrix}\begin{bmatrix} 1&0\\0&1\end{bmatrix} \begin{bmatrix} 1&0&0\\0&1&0\end{bmatrix} =\begin{bmatrix} 1&0&0\\ 0&1&0\\0&0&0\end{bmatrix} $$ If you remove row and column 1, you get $$ \begin{bmatrix} 1&0\\0&0\end{bmatrix}, $$ with rank $1\ne 2=r$.

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  • $\begingroup$ Right, I should have specified the conditions on X. I updated the question with the clear conditions on X. The example you mentioned, won't fit that condition. Thanks $\endgroup$ – Safwan Hossain Apr 4 at 1:51

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