EP morphisms

The concept of an EP matrix is extended to a morphism of a category C with involution. It is shown that an EP morphism has a group inverse iff it has a Moore-Penrose inverse, and in this case the inverses are identical. On the other hand, if a morphism has a Moore-Penrose inverse that is a group inverse, then C is a full subcategory of a category in which φ is EP. Also, if C is an additive category with involution 1 and with 1-biproduct factorization, then a morphism of φ of C is EP iff there is a 1-biproduct J ⊕ K and an invertible morphism θ : J → J such that φ is congruent to a morphism of the form

$\text{θ 0}\text{0 0}\text{: J⊕K → J⊕K.}$

In particular, a square matrix over a principal-ideal domain with involution is EP iff it is congruent to a matrix of the form dg(θ, 0) with θ invertible.