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  "body": "\n## Definition\n\nLet $X = (X_1, \\dots, X_n)^{\\mathsf{T}}$ be a random vector in $\\mathbb{R}^n$ with $E[X_i^2] \u003c \\infty$ for each $i$, and let $\\mu = E[X]$ be its mean vector.\nThe covariance matrix of $X$ is the $n \\times n$ matrix $\\Sigma = \\operatorname{Cov}(X)$ with entries\n$$\\Sigma_{ij} = \\operatorname{Cov}(X_i, X_j), \\quad 1 \\le i, j \\le n.$$\nIn matrix form, $\\Sigma = E\\left[(X - \\mu)(X - \\mu)^{\\mathsf{T}}\\right]$.\n\n## Remarks\n\nThe diagonal entries are the [[variances|007QED3E0Y]] $\\Sigma_{ii} = \\operatorname{Var}(X_i)$.\n$\\Sigma$ is symmetric and positive semidefinite: $a^{\\mathsf{T}} \\Sigma a = \\operatorname{Var}(a^{\\mathsf{T}} X) \\ge 0$ for each $a \\in \\mathbb{R}^n$.\nFor a real $m \\times n$ matrix $A$ and $b \\in \\mathbb{R}^m$, $\\operatorname{Cov}(AX + b) = A \\Sigma A^{\\mathsf{T}}$.\nThe covariance matrix and the mean vector are the two parameters of the [[multivariate normal distribution|0G558GJQMW]].\n\n## Explanations\n\n### L1 human\n\nFor a random vector, one number is not enough to describe the spread.\nThe covariance matrix collects the variance of each component and the covariance of each pair in one table.\n",
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