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Givens rotation
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===QR iteration variant=== If performing the above calculations as a step in the [[QR algorithm]] for finding the eigenvalues of a matrix, then one next wants to compute the matrix <math> RQ </math>, but one should not do so by first multiplying <math> G_{1}^T </math> and <math> G_{2}^T </math> to form <math> Q </math>, instead rather by multiplying each <math> G_k^T </math> by <math> R G_1^T \dots G_{k-1}^T </math> (on the right). The reason for this is that each multiplication by a Givens matrix on the right changes only two columns of <math> R </math>, thus requiring a mere <math> O(n) </math> arithmetic operations, which for <math> n-1 </math> Givens rotations sums up to <math> O(n^2) </math> arithmetic operations; multiplying by the general <math> n \times n </math> matrix <math>Q</math> would require <math> O(n^3) </math> arithmetic operations. Likewise, storing the full <math>Q</math> matrix amounts to <math>n^2</math> elements, but each Givens matrix is fully specified by its pair <math>(c,s)</math> and <math> n-1 </math> of them can thus be stored in <math> 2n-2 </math> elements. In the example, <math display="block"> \begin{aligned} R Q = A_3 (G_1^T G_2^T) ={}& (A_3 G_1^T) G_2^T \\ \approx{}& \begin{bmatrix} 8.8687& -1.5575& 2.5607\\ 2.9972& 3.5965& 0.9665\\ 0& 0& -4.1843 \end{bmatrix} G_2^T \approx \begin{bmatrix} 8.8687& 2.9972& 0.0\\ 2.9972&-1.0430&-3.5750\\ 0& -3.5750& 2.1742 \end{bmatrix} \end{aligned} </math>
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