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

Abstract

The Palais matrix represents an nn-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.

Keywords

• palais matrix
• complex householder transform
• unitary transform
• backward error analysis
• 15b10