Householder-Based QR and Singular Value Decompositions of Reduced Biquaternion Matrices, with an Application to Seismic Signal Denoising

Reduced biquaternion (RB) matrices provide a commutative algebraic framework for coupled multicomponent data, but their factorizations are usually formulated through complex or real representation matrices. The aim of this study is to develop and validate direct Householder-based QR and singular value decomposition (SVD) algorithms for RB matrices that operate in the native e1–e2 idempotent coordinates. The formulation treats vanishing idempotent components explicitly so that zero divisors require no invertibility assumption, keeps all factors in native RB form, and exposes two independent complex branches that execute concurrently. We distinguish implicit Householder reference constructions from accelerated LAPACK backends, establish their exact arithmetic equivalence, and derive normwise error bounds inherited from the complex kernels. An optional one-step residual-monotone correction lowers the measured reconstruction residual without altering the factorization invariants. Controlled benchmarks against real quaternion and commutative quaternion implementations, together with zero-divisor, rank-deficient, and ill-conditioned stress tests, confirm the roundoff-level residuals and competitive runtimes. The framework is then applied to three-component seismic denoising through a Hankel embedding and truncated RB-SVD, with parameters selected based on training noise realizations and evaluated on disjoint held-out realizations, including a reference-free rank selection variant and a noise level sensitivity study. The results establish a zero-divisor-safe and computationally efficient RB factorization framework for coupled multichannel low-rank processing.