This post describes how Maple was used to investigate the Givens rotation matrix, and to answer a simple question about its behavior. The "Givens" part is the medium, but the message is that it really is better to teach, learn, and do mathematics with a tool like Maple.
The question: If Givens rotations are used to take the vector Y = <5, -2, 1> to Y2 = , about what axis and through what angle will a single rotation accomplish the same thing?
The Givens matrix G21 takes Y to the vector Y1 =, and the Givens matrix G31 takes Y1 to Y2. Graphing the vectors Y, Y1, and Y2 reveals that Y1 lies in the xz-plane and that Y2 is parallel to the x-axis. (These geometrical observations should have been obvious, but the typical usage of the Givens technique to "zero-out" elements in a vector or matrix obscured this, at least for me.)
The matrix G = G31 G21 rotates Y directly to Y2; is the axis of rotation the vector W = Y x Y2, and is the angle of rotation the angle between Y and Y2? To test these hypotheses, I used the RotationMatrix command in the Student LinearAlgebra package to build the corresponding rotation matrix R. But R did not agree with G. I had either the axis or the angle (actually both) incorrect.
The individual Givens rotation matrices are orthogonal, so G, their product is also orthogonal. It will have 1 as its single real eigenvalue, and the corresponding eigenvector V is actually the direction of the axis of the rotation. The vector W is a multiple of <0, 1, 2> but V = <a, b, 1>, where . Clearly, W V.
The rotation matrix that rotates about the axis V through the angle isn't the matrix G either. The correct angle of rotation about V turns out to be
the angle between the projections of Y and Y2 onto the plane orthogonal to V. That came as a great surprise, one that required a significant adjustment of my intuition about spatial rotations. So again, the message is that teaching, learning, and doing mathematics is so much more effective and efficient when done with a tool like Maple.
A discussion of the Givens rotation, and a summary of the actual computations described above are available in the attached worksheet, What Gives with Givens.mw.