A generalization of the complex Autonne–Takagi factorization to quaternion matrices
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چکیده
This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. A complex symmetric matrix A can always be factored as A ¼ UAEU T , in which U is complex unitary and AE 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. 1. Introduction In 1915, Autonne [1] published a comprehensive original study of what would later be called the singular value decomposition of complex matrices, with special attention to uniqueness of the factors. As applications of his results, he obtained special singular value decompositions for symmetric, coninvolutory, normal, orthogonal and Lorentzian matrices. His special singular value decomposition for a complex symmetric matrix A is
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