Factorization of matrices of quaternions
Copyright policy )Factorization of matrices of quaternions
We review known factorization results in quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, the QR factorization. We prove there is a Schur factorization for commuting matrices, and from this derive the spectral theorem. We do not consider algorithms, but do point to some of the numerical literature. Rather than work directly with matrices of quaternions, we work with complex matrices with a specific symmetry based on the dual operation. We discuss related results regarding complex matrices that are self-dual or symmetric, but perhaps not Hermitian.
Corrected proofs of Theorem 2.4(2) and Theorem 3.2
- University of New Mexico United States
quaternions, Mathematics(all), Eigenvalues, singular values, and eigenvectors, Matrices over special rings (quaternions, finite fields, etc.), Canonical forms, reductions, classification, dual operation, singular value decomposition, QR factorization, Mathematics - Operator Algebras, 15B33, spectral theorem, Factorization of matrices, Kramers pair, Kramers degeneracy, polar decomposition, matrix decompositions, Matrix decompositions, Jordan canonical form, FOS: Mathematics, Quaternions, Operator Algebras (math.OA), Dual operation, Schur factorization
quaternions, Mathematics(all), Eigenvalues, singular values, and eigenvectors, Matrices over special rings (quaternions, finite fields, etc.), Canonical forms, reductions, classification, dual operation, singular value decomposition, QR factorization, Mathematics - Operator Algebras, 15B33, spectral theorem, Factorization of matrices, Kramers pair, Kramers degeneracy, polar decomposition, matrix decompositions, Matrix decompositions, Jordan canonical form, FOS: Mathematics, Quaternions, Operator Algebras (math.OA), Dual operation, Schur factorization
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