Find rank of matrix

Question

If $X$ is a $N \times D$ matrix with $(D\gg N)$ with $\operatorname{rank}(X) = N$, what is $\operatorname{rank}(X^T \cdot X)$ where $X^T$ is the transpose matrix of $X$?

I am little new to linear algebra and I am not having any approach, I faced this problem in other context(linear regression).

https://math.stackexchange.com/questions/2315/is-the-rank-of-a-matrix-the-same-of-its-transpose-if-yes-how-can-i-prove-it — Stockfish, Jul 21 '20 at 14:58
Welcome to MSE. When asking questions, people will respond more positively if you: list your efforts and thoughts and type up your question using MathJax instead of linking images. Please consider editing your question. — Integrand, Jul 21 '20 at 15:01
@Stockfish The post you linked isn't obviously relevant, and there definitely are posts on MSE about the rank of $X^TX$ — Ben Grossmann, Jul 21 '20 at 15:17
@BenGrossmann Oh, will look for proof. If you have any link handy, please share. — illa, Jul 21 '20 at 15:22

score 0 · Accepted Answer · answered Jul 21 '20 at 15:31

The rank of $X^TX$ is the same as the rank of $X$, assuming real matrices.

Why? Suppose $Xv=0$. Then also $X^TXv=0$. Conversely, if $X^TXv=0$, then also $v^TX^TXv=0$, that is, $(Xv)^T(Xv)=0$; therefore $Xv=0$.

We have proved that the two matrices have the same null space. But, by the rank-nullity theorem, \begin{align} N&=\operatorname{rank}X+\operatorname{nullity}X \\ N&=\operatorname{rank}X^TX+\operatorname{nullity}X^TX \end{align} and equality of the ranks follows.

Find rank of matrix

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