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  "body": "\n## Definition\n\nLet $D = (d_{ij})$ be an $n \\times n$ [[symmetric matrix|0VR671PHE5]] with $d_{ii} = 0$ and $d_{ij} \\ge 0$, the dissimilarities of $n$ objects, and let $1 \\le k \u003c n$.\n*Multidimensional scaling* seeks points $x_1, \\ldots, x_n \\in \\R^k$ whose distances $\\abs{x_i - x_j}$ approximate the $d_{ij}$.\n*Classical multidimensional scaling* computes them as follows:\n\n1. Let $D^{(2)} = (d_{ij}^2)$ and $J = I_n - \\tfrac{1}{n} \\mathbf{1} \\mathbf{1}^T$, where $\\mathbf{1}$ is the vector of ones, and let $B = -\\tfrac{1}{2} J D^{(2)} J$.\n2. Let $\\lambda_1 \\ge \\cdots \\ge \\lambda_k$ be the $k$ largest [[eigenvalues|0BQAA2BK3Q]] of $B$, assumed positive, with orthonormal [[eigenvectors|0N6960W0BV]] $e_1, \\ldots, e_k$.\n3. Let $x_i \\in \\R^k$ be the $i$-th row of the $n \\times k$ matrix $\\begin{pmatrix} \\sqrt{\\lambda_1} e_1 \u0026 \\cdots \u0026 \\sqrt{\\lambda_k} e_k \\end{pmatrix}$.\n\n## Remarks\n\nThe matrix $D$ is the matrix of [[Euclidean distances|0NPPGHQNXQ]] of some points in $\\R^k$ exactly when $B$ is [[positive semidefinite|01MMVW8CKC]] of rank at most $k$.\nIn that case classical scaling recovers the points up to a translation and an orthogonal transformation, since $B$ is then the Gram matrix of the centred points.\nExample: three points on a line at distances $d_{12} = d_{23} = 1$, $d_{13} = 2$.\nThe centred configuration $-1, 0, 1$ has Gram matrix $B = \\begin{pmatrix} 1 \u0026 0 \u0026 -1 \\\\ 0 \u0026 0 \u0026 0 \\\\ -1 \u0026 0 \u0026 1 \\end{pmatrix}$, with largest eigenvalue $2$ and eigenvector $(1, 0, -1)/\\sqrt{2}$.\nStep 3 gives the coordinates $1, 0, -1$, the original points after a reflection.\nNon-example: if $d_{12} = d_{23} = 1$ and $d_{13} = 3$, the triangle inequality fails, so no points in any $\\R^k$ realise $D$, and $B$ has a negative eigenvalue.\nMetric and non-metric scaling instead minimise a stress function, such as $\\sum_{i \u003c j} (d_{ij} - \\abs{x_i - x_j})^2$, by iteration.\n",
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