Real matrix homotopies based on Householder reflections

Anexplicithomotopyisconstructedbetweentheidentitymatrixandanarbitraryrealmatrixwithpositivedeterminant.Theconstructionrequiresnoeigenvalueinformationandyieldsasmoothparameterizationforwhichthematrixdeterminantremainspositiveandvariesas a monotonic or unimodal function of the parameter. The key idea of the construction is that a simple formula smoothly transforms any product of two Householder reflections to the identity.