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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $D = (d_{ij})$ be a real symmetric $n \\times n$ matrix with $d_{ii} = 0$ for all $i$.\nLet $G$ be the $(n - 1) \\times (n - 1)$ matrix with entries\n$$G_{ij} = \\frac{1}{2} \\left( d_{1i} + d_{1j} - d_{ij} \\right), \\qquad 2 \\le i, j \\le n.$$\nThen the following hold.\n\n1. $D$ is a [[Euclidean distance matrix|1BCJ7A73KS]] if and only if $G$ is [[positive semidefinite|01MMVW8CKC]].\n2. In that case, points of $\\R^k$ realize $D$ if and only if $k$ is at least the [[rank|0ZC9YTXZRY]] of $G$.\n\n## Remarks\n\nIf $x_1, \\ldots, x_n$ realize $D$, translate so that $x_1 = 0$.\nThe polarization identity $\\ip{x_i}{x_j} = \\frac{1}{2}(\\norm{x_i}^2 + \\norm{x_j}^2 - \\norm{x_i - x_j}^2)$ shows that $G$ is the [[Gram matrix|00SPJSDT3Z]] of $x_2, \\ldots, x_n$, hence positive semidefinite.\nConversely, a positive semidefinite $G$ of rank $k$ factors as $G = X^T X$ with $X$ a $k \\times (n - 1)$ matrix.\nIts columns, together with $x_1 = 0$, realize $D$ in $\\R^k$.\n\nExample: the matrix with rows $(0, 1, 9)$, $(1, 0, 1)$ and $(9, 1, 0)$ gives $G = \\begin{pmatrix} 1 \u0026 4.5 \\\\ 4.5 \u0026 9 \\end{pmatrix}$.\nIts determinant is $9 - 20.25 \u003c 0$, so $G$ is not positive semidefinite and $D$ is not a Euclidean distance matrix.\n\nAn equivalent form uses the centring matrix $J = I - \\frac{1}{n} \\mathbf{1}\\mathbf{1}^T$: $D$ is a Euclidean distance matrix if and only if $-\\frac{1}{2} J D J$ is positive semidefinite.\n\n## Explanations\n\n### L1 human\n\nA table of squared distances comes from real points exactly when a related matrix has no negative eigenvalues.\nThis turns a geometric question into a check that a computer does quickly.\nThe number of nonzero eigenvalues tells you how many dimensions the points need.\n\n### L2 human\n\nThe criterion is the basis of classical multidimensional scaling.\nOne forms $-\\frac{1}{2} J D J$, takes its eigendecomposition, and reads off coordinates from the top eigenvectors.\nIt is the matrix side of [[distance geometry|HPW5F0GQ4B]], and it is closely related to the [[Cayley–Menger determinant|KGC5CYDF53]].\nThe criterion is named after Isaac Schoenberg.\n",
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