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  "body": "\n## Summary\n\nA Givens rotation is an $n \\times n$ [[orthogonal matrix|0W02QESPJQ]] that [[rotates|0HFREYRSZP]] the plane of two coordinate axes and leaves all other coordinates fixed.\nWith the angle chosen well, it turns one chosen entry of a vector or matrix into zero.\nBecause it touches only two rows, it is the tool of choice for zeroing single entries.\nThis is used in the [[QR decomposition|0ERWG4NXQ9]] of sparse or structured matrices and in the QR algorithm for eigenvalues.\n\n## Explanations\n\n### L1 human\n\nPick two coordinates, say the first and the third, and turn the vector inside the plane they span, keeping every other coordinate as it is.\nTurning by a suitable angle makes one of the two coordinates zero, and moves all the length into the other one.\nFor example, rotating $(3, 4)$ clockwise by the angle whose cosine is $3/5$ gives $(5, 0)$.\nDoing this entry by entry clears a matrix below its diagonal.\n",
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