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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $1 \\le i \u003c j \\le n$, and let $c, s$ be real numbers with $c^2 + s^2 = 1$.\nThe *Givens rotation* $G(i, j; c, s)$ is the $n \\times n$ real matrix that agrees with the [[identity matrix|06DW9BAWB1]] except in the four entries\n$$g_{ii} = c, \\qquad g_{ij} = -s, \\qquad g_{ji} = s, \\qquad g_{jj} = c.$$\nWriting $c = \\cos\\theta$ and $s = \\sin\\theta$, it is the [[rotation|0XNZWJ96HK]] by the angle $\\theta$ in the plane of the coordinates $i$ and $j$.\n\n## Remarks\n\n$G = G(i, j; c, s)$ is an [[orthogonal matrix|0QRBVFJ50X]] with determinant $c^2 + s^2 = 1$.\nFor $x \\in \\R^n$, the product $G x$ changes only the entries $i$ and $j$: $(Gx)_i = c x_i - s x_j$ and $(Gx)_j = s x_i + c x_j$.\nIf $(x_i, x_j) \\ne (0, 0)$, put $r = \\sqrt{x_i^2 + x_j^2}$, $c = x_i / r$ and $s = -x_j / r$; then $(Gx)_i = r$ and $(Gx)_j = 0$.\nExample: for $x = (3, 4)$ in $\\R^2$, $r = 5$, $c = 3/5$, $s = -4/5$, and $G x = (9/5 + 16/5, \\ -12/5 + 12/5) = (5, 0)$.\nNon-example: the matrix with $c = s = 1$ in these four places is not a Givens rotation, since $c^2 + s^2 = 2$, and it is not orthogonal.\nOther books put $-s$ in position $(j, i)$ instead; the two conventions differ by the sign of $\\theta$.\n\n## Explanations\n\n### L1 human\n\nA Givens rotation turns a vector inside the plane of two coordinate axes and does nothing to the other coordinates.\nIts matrix is the identity with a $2 \\times 2$ rotation block written into rows and columns $i$ and $j$.\nThe angle can be chosen so that one of the two entries becomes zero.\n",
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