In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes. Givens rotations are named after Wallace Givens, who introduced them to numerical analysts in the 1950s while he was working at Argonne National Laboratory.
As action on matrices A Givens rotation acting on a matrix from the left is a row operation, moving data between rows but always within the same column. Unlike the elementary operation of row-addition, a Givens rotation changes both of the rows addressed by it. To understand how it is a rotation, one may denote the elements of one target row by x 1 {\displaystyle x_{1}} through x n {\displaystyle x_{n}} and the elements of the other target row by y 1 {\displaystyle y_{1}} through y n {\displaystyle y_{n}} :
[ ⋮ ⋮ ⋱ ⋮ x 1 x 2 … x n ⋮ ⋮ ⋱ ⋮ y 1 y 2 … y n ⋮ ⋮ ⋱ ⋮ ] {\displaystyle {\begin{bmatrix}\vdots &\vdots &\ddots &\vdots \\x_{1}&x_{2}&\dots &x_{n}\\\vdots &\vdots &\ddots &\vdots \\y_{1}&y_{2}&\dots &y_{n}\\\vdots &\vdots &\ddots &\vdots \end{bmatrix}}}
Then the effect of a Givens rotation is to rotate each subvector ( x k , y k ) {\displaystyle (x_{k},y_{k})} by the same angle. As with row-addition, algorithms often choose this angle so that one specific element becomes zero, and whatever happens in remaining columns is considered acceptable side-effects. A Givens rotation acting on a matrix from the right is instead a column operation, moving data between two columns but always within the same row. As with action from the left, it rotates each subvector ( x k , y k ) {\displaystyle (x_{k},y_{k})} by the same angle, but here these named elements occur in the matrix as
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