Two-Dimensional Givens Rotation
The following matrix is the two-dimensional Givens Rotation from the axis to the axis:
This matrix rotates two-dimensional column vectors about the origin of the – plane.
For instance, if we rotate the unit column vector using this Givens Rotation, then we have:
Therefore, if we rotate the column vector , then we have:
and to rotate that column vector back to the axis so that the value is zero, we would have:
So, we need to use an angle such that the value goes to zero. Let’s concentrate on that lower equation:
atan2( y, x ) in C/C++ and
ArcTan[ x, y ] in Mathematica.
Therefore, when we rotate the column vector back to the axis so that the value is zero, we have:
Since the radius from the origin to the column vector is (also known as the norm), then we also have:
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