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  "body": "\n## Definition\n\nLet $X$ be a random vector in $\\R^p$ and $Y$ a random vector in $\\R^q$, with finite second moments.\nAssume that the [[covariance matrices|0V7G6WMFK0]] $\\Sigma_{XX}$ and $\\Sigma_{YY}$ are [[positive definite|0KX66CZMBB]], and let $\\Sigma_{XY} = \\operatorname{Cov}(X, Y)$.\nLet $m = \\min(p, q)$.\nFor $i = 1, \\ldots, m$, the $i$-th pair of *canonical variates* $(a_i^T X, b_i^T Y)$ is defined inductively:\n$a_i \\in \\R^p$ and $b_i \\in \\R^q$ maximise the [[correlation|0DNN9B6Q2F]] $\\operatorname{Corr}(a^T X, b^T Y)$ under these constraints:\n\n1. $\\operatorname{Var}(a^T X) = \\operatorname{Var}(b^T Y) = 1$;\n2. $\\operatorname{Cov}(a^T X, a_j^T X) = 0$ for all $j \u003c i$;\n3. $\\operatorname{Cov}(b^T Y, b_j^T Y) = 0$ for all $j \u003c i$.\n\nThe maximal value $\\rho_i = \\operatorname{Corr}(a_i^T X, b_i^T Y)$ is the $i$-th *canonical correlation*.\n\n## Remarks\n\nOne has $1 \\ge \\rho_1 \\ge \\rho_2 \\ge \\cdots \\ge \\rho_m \\ge 0$.\nThe numbers $\\rho_1^2, \\ldots, \\rho_m^2$ are the $m$ largest [[eigenvalues|0BQAA2BK3Q]] of $\\Sigma_{XX}^{-1} \\Sigma_{XY} \\Sigma_{YY}^{-1} \\Sigma_{YX}$.\nExample: for $p = q = 1$, the only canonical correlation is $\\rho_1 = \\abs{\\operatorname{Corr}(X, Y)}$, attained with $b_1 = \\pm 1/\\sqrt{\\operatorname{Var}(Y)}$ chosen with the sign of the covariance.\nNon-example: for $X = (X_1, X_2)$ with $X_1 = Y$ and $X_2$ independent of $Y$, the pair $a = (0, 1)/\\sqrt{\\operatorname{Var}(X_2)}$ gives correlation $0$, which is not maximal, since $a = (1, 0)/\\sqrt{\\operatorname{Var}(Y)}$ gives correlation $1$.\nThe sample version replaces the covariance matrices by sample covariance matrices.\n",
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