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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nLet $D = (d_{ij})$ be a real $n \\times n$ matrix.\nIt is a *Euclidean distance matrix* if some integer $k \\ge 1$ and points $x_1, \\ldots, x_n \\in \\R^k$ satisfy\n$$d_{ij} = \\norm{x_i - x_j}^2 \\quad \\text{for all } i, j,$$\nwhere $\\norm{x_i - x_j}$ is the [[Euclidean distance|0C95TG69SG]] of $x_i$ and $x_j$.\nThe points $x_1, \\ldots, x_n$ are then said to realize $D$.\n\n## Remarks\n\nEvery Euclidean distance matrix is symmetric, has zero diagonal and has nonnegative entries.\n\nExample: the points $0, 1, 3 \\in \\R$ give\n$$D = \\begin{pmatrix} 0 \u0026 1 \u0026 9 \\\\ 1 \u0026 0 \u0026 4 \\\\ 9 \u0026 4 \u0026 0 \\end{pmatrix},$$\nsince $1^2 = 1$, $3^2 = 9$ and $2^2 = 4$.\n\nNon-example: the matrix with rows $(0, 1, 9)$, $(1, 0, 1)$ and $(9, 1, 0)$ is symmetric with zero diagonal, but it is not a Euclidean distance matrix.\nPoints realizing it would have distances $1$, $1$ and $3$, and $3 \u003e 1 + 1$ violates the triangle inequality.\n\nSome authors use the distances themselves instead of their squares.\n",
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