Full rank decomposition-based representations for quaternion generalized inverses

Develop full rank decomposition-based representations for outer inverses and {1,2}-inverses of quaternion matrices with prescribed range and/or null space constraints, providing explicit formulas analogous to those known for complex matrices.

Background

Full rank decomposition techniques yield explicit representations for generalized inverses in the complex case, including inverses constrained by subspace properties. Extending these techniques to quaternion matrices is complicated by the separate treatment of left and right spaces and non-commutativity.

The authors explicitly identify this extension as an open question in the introduction; later sections of the paper present decomposition-based results, but the quoted text records the problem as originally posed.