$X\succeq0$ with rank $r\leq n\Rightarrow X=V^TV$ where $V\in\mathbb{R}^{r\times n}$

Question

I saw this statement, and was wondering about the proof:

$X\in\mathbb{R}^{n\times n}$ is positive semidefinite with rank $r\Longrightarrow X=V^TV$ where $V\in\mathbb{R}^{r\times n}$.

Answer 1

Hint: Take $V$ to be the matrix whose $i$th row is $\sqrt{\lambda_i} v_i$, where $\lambda_i$ is the $i$th eigenvalue of $X$.