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  "body": "\n## Definition\n\nLet $X$ be a random vector in $\\R^p$ with [[mean|0WQNRCNY3M]] $\\mu$ and [[covariance matrix|0V7G6WMFK0]] $\\Sigma$, and assume that $\\Sigma$ is [[positive definite|0KX66CZMBB]].\nThe *Mahalanobis distance* of a point $x \\in \\R^p$ from the distribution of $X$ is\n$$D_M(x) = \\sqrt{(x - \\mu)^T \\Sigma^{-1} (x - \\mu)}.$$\nThe Mahalanobis distance between two points $x, y \\in \\R^p$ with respect to $\\Sigma$ is\n$$d_M(x, y) = \\sqrt{(x - y)^T \\Sigma^{-1} (x - y)}.$$\n\n## Remarks\n\nExample: let $\\mu = 0$ and $\\Sigma = \\operatorname{diag}(4, 1)$.\nThen $\\Sigma^{-1} = \\operatorname{diag}(1/4, 1)$, so $D_M(2, 0) = \\sqrt{4/4} = 1$ and $D_M(0, 1) = 1$, while $D_M(0, 2) = 2$.\nNon-example: if $\\Sigma = \\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$, then $\\Sigma$ is not invertible, so the formula is not defined.\nIf $\\Sigma = A A^T$ with $A$ invertible, then $d_M(x, y) = \\abs{A^{-1} x - A^{-1} y}$, the Euclidean distance after the change of variables $x \\mapsto A^{-1} x$.\nSo $d_M$ is a metric on $\\R^p$.\nIf $\\Sigma$ is the identity, $d_M$ is the Euclidean distance.\nIf $X$ is multivariate normal, then $D_M(X)^2$ has the chi-squared distribution with $p$ degrees of freedom.\n",
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