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  "body": "\n## Statement\n\nLet $A \\in \\R^{n \\times n}$, $n \\ge 2$, be [[symmetric|0VR671PHE5]], and let $\\mathrm{off}(A)^2 = \\sum_{i \\ne j} a_{ij}^2$.\nFor indices $p \u003c q$ with $a_{pq} \\ne 0$, let $J = J(p, q, \\theta)$ be the rotation in the $(p, q)$ coordinate plane, where $\\theta \\in [-\\pi/4, \\pi/4]$ solves $\\cot 2\\theta = (a_{qq} - a_{pp})/(2 a_{pq})$.\nThen $A' = J^T A J$ is symmetric, has the same [[eigenvalues|0BQAA2BK3Q]] as $A$, satisfies $a'_{pq} = 0$, and\n$$\\mathrm{off}(A')^2 = \\mathrm{off}(A)^2 - 2 a_{pq}^2.$$\nThe *classical Jacobi method* chooses at each step a pair $(p, q)$ with $\\abs{a_{pq}}$ maximal among the off-diagonal entries and sets $A_{k+1} = J_k^T A_k J_k$, starting from $A_0 = A$.\nThen\n$$\\mathrm{off}(A_{k+1})^2 \\le \\Bigl( 1 - \\frac{2}{n(n-1)} \\Bigr) \\mathrm{off}(A_k)^2,$$\nso $A_k$ converges to a [[diagonal matrix|044JNC7ZAA]] whose diagonal entries are the eigenvalues of $A$.\n\n## Remarks\n\nThe rotation $J$ is an [[orthogonal matrix|0W02QESPJQ]], so $A'$ is [[similar|0E89HQATM8]] to $A$ and the [[Frobenius norm|JT8Q204E4R]] $\\norm{A'}_F = \\norm{A}_F$ is unchanged.\nOn the $2 \\times 2$ block in rows and columns $p, q$, the choice of $\\theta$ diagonalizes the block.\nSo the sum of squares of that block's diagonal grows by $2 a_{pq}^2$.\nThe other entries in rows and columns $p, q$ are only rotated, so their total square sum does not change.\nIf $\\abs{a_{pq}}$ is maximal among the $n(n-1)$ off-diagonal entries, then $a_{pq}^2 \\ge \\mathrm{off}(A)^2 / (n(n-1))$, which gives the linear bound.\nThe diagonal entries of $A_k$ approach the eigenvalues because each eigenvalue lies within $\\mathrm{off}(A_k)$ of some diagonal entry, by Weyl's inequality.\n\nExample: for $A = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 2 \\end{pmatrix}$, $\\cot 2\\theta = 0$ gives $\\theta = \\pi/4$, and one rotation gives $\\mathrm{diag}(1, 3)$.\nIn practice the local convergence is quadratic once the off-diagonal part is small.\nThe product $J_0 J_1 \\cdots J_k$ of the rotations converges to an orthogonal matrix whose columns are eigenvectors.\n\n## Explanations\n\n### L1 human\n\nThe Jacobi method diagonalizes a symmetric matrix by a sequence of plane rotations.\nEach rotation kills one off-diagonal entry; later rotations may make it nonzero again, but the total size of the off-diagonal part always drops.\nAfter enough rotations the matrix is nearly diagonal and the eigenvalues can be read off the diagonal.\n\n### L2 human\n\nThe method dates from Jacobi in 1846 and was revived on early computers.\nIt is slower than the [[QR algorithm|B6F8YWHYZ5]] on a serial computer.\nBut it computes small eigenvalues of well-scaled positive definite matrices to high relative accuracy, and it is easy to parallelize.\nCyclic variants sweep through all pairs in a fixed order instead of searching for the largest entry, and they also converge.\nA one-sided version applied to $A^T A$ implicitly computes the [[singular value decomposition|000NPTMX6J]].\n",
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