Result: A note on hyperbolic transformations: A note on hyperbolic transformations.

The well known Givens (GT) and Householder (HT) transformations are used in matrix factorization. They are related to the identity matrix \(I\), e.g., \(F^HIF=I\), where \(F^H\) stands for the Hermitian transpose of the GT matrix \(F\). The hyperbolic counterparts to the GT and HT (HGT and HHT for short) are related to matrices \(\Phi =\operatorname {diag}(\pm 1,\dots ,\pm 1)\) appearing in some singular value decomposition problems. It holds \(M^H\Phi M=\Phi \), where \(M\) is an HGT or HHT matrix. If \(\Phi =I\), then the HGT and HHT come down to the GT and HT, respectively. The authors give a thorough analysis of basic properties of the HGT and HHT applied to complex matrices, including the case of zero hyperbolic energy of a transformed vector, i.e., a real number dependent on \(\Phi \) which becomes a common norm of a vector if \(\Phi =I\). Relevant MATLAB algorithms are available at \texttt{http://www.math.uni-hamburg.de/home/opfer/veroeffentlichungen.html}.