Complex Palais matrix and a new unitary transform with bounded component norms

The Palais matrix represents an -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 transform, which encompasses the transform by the Palais matrix, the Householder reflection, and their unitary extensions. Furthermore, we define the , a variant of the transform featuring bounded component norms. It is demonstrated that the 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 transform exhibits additional characteristics, including its convergence to the identity and the rowwise structure of its backward error.