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 factorizationcomplex symmetric matrixquaternion matrixsingular value decompositioncanonical formsAMS Subject Classifications:: 15A2315A33 Acknowledgements It is a pleasure to acknowledge an insightful comment from Tatiana Klimchuk that helped us improve the formulation of Theorem 3.