A Nonzero Positive Semidefinite Matrix Factors as BBᵀ with BᵀB Invertible

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $d\in\mathbb{N}$ and let $\Sigma\in\mathcal{S}(d)$ be positive semidefinite with $\Sigma\neq0_{d}$.

(i) There are $r\in[d]$ and $B\in\mathbb{R}^{d\times r}$ such that $B^{\top}B$ is invertible and

$$BB^{\top}=\Sigma.$$