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LU Decomposition and Generalized Autoone-Takagi Decomposition of Dual Matrices and their Applications

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arxiv 2501.04324 v1 pith:KADYIYCW submitted 2025-01-08 math.NA cs.NA

LU Decomposition and Generalized Autoone-Takagi Decomposition of Dual Matrices and their Applications

classification math.NA cs.NA
keywords decompositiondualmatricesapplicationsautoone-takagigeneralizedsymmetricalgebra
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abstract

This paper uses matrix transformations to provide the Autoone-Takagi decomposition of dual complex symmetric matrices and extends it to dual quaternion $\eta$-Hermitian matrices. The LU decomposition of dual matrices is given using the general solution of the Sylvester equation, and its equivalence to the existence of rank-k decomposition and dual Moore-Penrose generalized inverse (DMPGI) is proved. Similar methods are then used to provide the Cholesky decomposition of dual real symmetric positive definite matrices. Both of our decompositions are driven by applications in numerical linear algebra.

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