抄録
Abstract
The Palais matrix represents an n n -dimensional rotation between two vectors that is functionally equivalent to the Householder reflection. This study introduces a one-parameter family of unitary transforms, termed the θ \theta transform, which encompasses the transform by the Palais matrix, the Householder reflection, and their unitary extensions. Furthermore, we define the θ ∠ {\theta }_{\angle } transform, a variant of the θ \theta transform featuring bounded component norms. It is demonstrated that the θ ∠ {\theta }_{\angle } transform is computationally efficient and backward stable when one of the vectors has the “one-hot” structure, making it highly valuable for matrix decompositions such as the QR decomposition. In addition, the θ \theta transform exhibits additional characteristics, including its convergence to the identity and the rowwise structure of its backward error.