In "The Theory of Matrices" by Gantmacher, Chapter II, section 4:

Corollary $3$: If $S= ||s_{ik}||_1^n$ is a symmetric matrix of rank $r$ and $$D_k= S\bigl( \begin{smallmatrix} 1 & 2 & \dots & k \\ 1 & 2 & \dots & k \end{smallmatrix} \bigr)\ne 0 \ \ \ \ \ \ \ (k= 1,2, \dots,r),$$

then $$S=BB'$$ where $B=||b_{ik}||_1^n$ is a lower triangular matrix in which $$b_{gk}= \begin{Bmatrix} \frac{1}{\sqrt{D_kD_{k-1}}} S\bigl( \begin{smallmatrix} 1 & 2 & \dots & k-1 & g \\ 1 & 2 & \dots & k-1 & k \end{smallmatrix} \bigr) & (g=k, k+1, \dots, n; k= 1,2, \dots, r) \\ 0 & (g=k, k+1, \dots, n; k= r+1, \dots, n) \\ \end{Bmatrix}$$

The question:
How can one ensure that $D_kD_{k-1}$ is always greater than $0$?