Iterative Methods for Computing the Moore-Penrose Pseudoinverse of Quaternion Matrices, with Applications

We develop quaternion–native iterative methods for computing the Moore–Penrose (MP) pseudoinverse of quaternion matrices and analyze their convergence. Our starting point is a damped Newton–Schulz (NS) iteration tailored to noncommutativity: we enforce the appropriate left/right identities for rectangular inputs and prove convergence directly in $\mathbb{H}$ under a simple spectral scaling. We then derive higher–order (\emph{hyperpower}) NS schemes with exact residual recurrences that yield order-$p$ local convergence, together with factorizations that reduce the number of $s\times s$ quaternion products per iteration. Beyond NS, we introduce a randomized sketch–and–project method (RSP–Q), a hybrid RSP+NS scheme that interleaves inexpensive randomized projections with an exact hyperpower step, and a matrix–form conjugate gradient on the normal equations (CGNE–Q). All algorithms operate directly in $\mathbb{H}$ (no real or complex embeddings) and are matrix–free.