A generalization of the complex Autonne–Takagi factorization to quaternion matrices

Abstract

A complex symmetric matrix A can always be factored as A = UΣU ^T , in which U is complex unitary and Σ is a real diagonal matrix whose diagonal entries are the singular values of A. This factorization may be thought of as a special singular value decomposition for complex symmetric matrices. We present an analogous special singular value decomposition for a class of quaternion matrices that includes complex matrices that are symmetric or Hermitian.

Keywords:

• Autonne–Takagi factorization
• complex symmetric matrix
• quaternion matrix
• singular value decomposition
• canonical forms

AMS Subject Classifications::

• 15A23
• 15A33

Acknowledgements

It is a pleasure to acknowledge an insightful comment from Tatiana Klimchuk that helped us improve the formulation of Theorem 3.