Extend Urquhart representation to quaternion matrices

Establish an extension of Urquhart’s representation for generalized inverses with prescribed range and/or null space constraints from complex matrices to matrices over the quaternion skew field, taking into account the non-commutativity and the distinction between left and right quaternion vector spaces.

Background

Urquhart’s representation provides explicit formulas for generalized inverses (with prescribed range and null space) in the real and complex settings. Extending such representations to quaternion matrices is nontrivial due to the non-commutative nature of quaternions and the need to treat left and right range/null spaces separately.

The paper highlights this as an open question motivating their work. The authors later present results aiming to provide Urquhart-type representations for quaternion generalized inverses, but the quoted passage explicitly frames the problem as open at the outset.