LAUSR

Using the concept of a transposition anti-involution in Clifford algebra $$C \, \ell _{1,1}$$ C ℓ 1 , 1 and the isomorphisms $${\mathbb {H}}_s \cong C \, \ell _{1,1} \cong \text {Mat}(2,{\mathbb {R}}),$$ H s ≅ C ℓ 1 , 1 ≅ Mat ( 2 , R ) , where $${\mathbb {H}}_s$$ H s is the algebra of split quaternions and $$\text {Mat}(2,{\mathbb {R}})$$ Mat ( 2 , R ) is the algebra of $$2 \times 2$$ 2 × 2 real matrices, one can find the Moore–Penrose inverse $$q^{+}$$ q + of a non-zero non-invertible split quaternion  q . In particular, using a well-known algorithm for finding the Moore-Penrose inverse $$Q^{+}$$ Q + of a non-zero $$2 \times 2$$ 2 × 2 matrix Q of rank 1, one can give four governing equations that the (unique) split quaternion $$q^{+}$$ q + corresponding to $$Q^{+}$$ Q + must satisfy. We show how a dyadic expansion and a Singular Value Decomposition can be found for any split quaternion q , and we relate them to $$q^{+}$$ q + . Results presented in this paper may be useful in a plethora of recent applications of split quaternions.