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Source. Published p. 218, Theorem 2.1, attributed there to Schoenberg (1938). (canonical PDF).

A symmetric real matrix $M=(m_{ij})_{i,j=1}^{d+1}$ with zero diagonal is of negative type if

$$\sum_{i=1}^{d}\sum_{j=i+1}^{d+1}m_{ij}\zeta_i\zeta_j\le0$$

for all $\zeta_1,\ldots,\zeta_{d+1}$ with $\sum_i\zeta_i=0$ and $\sum_i\zeta_i^2=1$ (inequality (1), p. 218). As printed, Theorem 2.1 states that a finite metric space $X=\{x_1,\ldots,x_{d+1}\}$ with distances $d_{ij}$ embeds in $\mathbb R^d$ if and only if the matrix with entries $m_{ij}=d_{ij}^2$ is of negative type, and that the embedded image is affinely independent if and only if inequality (1) is always strict.

Array form used here. Let $e_{ij}=e_{ji}$ be real numbers with $e_{ii}=0$, indexed by $1\le i,j\le n$. There are points $x_1,\ldots,x_n\in\mathbb R^{n-1}$ with $\|x_i-x_j\|^2=e_{ij}$ if and only if

$$Q_e(\lambda)=\sum_{i<j}e_{ij}\lambda_i\lambda_j\le0 \quad\text{for every }\lambda\text{ with }\sum_i\lambda_i=0.$$

The realization is affinely independent exactly when the inequality is strict for every nonzero such vector. By homogeneity it is enough to test $\sum_i\lambda_i^2=1$.

Proof pointer and scope. The complete elementary proof, including the semidefinite case, the anchored Gram construction, and a uniform strict negative margin, is already at negative_type_criterion. It is used here without duplicating that proof. The original 1938 paper's full proof is not claimed to have been reviewed. The printed theorem assumes a finite metric space. The array form above also permits coincident points in the non-strict case, and it does not require a prior metric realization.

Bears on. #174.