Abstract
Givens' transformation (1954) was originally applied to real matrices. We shall give an extension to quaternion valued matrices. The complex case will be treated in the introduction. We observe that the classical Givens' rotation in the real and in the complex case is itself a quaternion using an isomorphism between certain (2×2) matrices and R 4 equipped with the quaternion multiplication. In the real and complex case Givens' (2×2) matrix is determined uniquely up to an arbitrary (real or complex) factor σ with |σ|=1. However, because of the noncommutativity of quaternions, we shall show that in the quaternion case such a factor must obey certain additional restrictions. There are two numerical examples including a MATLAB program and some hints for implementation. Matrices with quaternion entries arise, e.g., in quantum mechanics.
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