About the von Neumann regularity of triangular block matrices

Necessary and sufficient conditions are given for the von Neumann regularity of triangular block matrices with von Neumann regular diagonal blocks over arbitrary rings. This leads to the characterization of the von Neumann regularity of a class of triangular Toeplitz matrices over arbitrary rings. Some special results and a new algorithm are derived for triangular Toeplitz matrices over commutative rings. Finally, the Drazin invertibility of some companion matrices over arbitrary rings is considered, as an application.Fundação para a Ciência e a Tecnologia (FCT)

Generalized invertibility in two semigroups of a ring

In {\em Linear and Multilinear Algebra}, 1997, Vol.43, pp.137-150, R. Puystjens and R. E. Hartwig proved that given a regular element $t$ of a ring $R$ with unity $1$, then $t$ has a group inverse if and only if $u=t^{2}t^{-}+1-tt^{-}$ is invertible in $R$ if and only if $v=t^{-}t^{2}+1-t^{-}t$ is invertible in $R$. There, R. E. Hartwig posed the pertinent question whether the inverse of $u$ and $v$ could be directly related. Similar equivalences appear in the characterization of Moore-Penrose and Drazin invertibility, and therefore analogous questions arise. We present a unifying result to answer these questions not only involving classical invertibility, but also some generalized inverses as well.Fundação para a Ciência e a Tecnologia (FCT) - Programa Operacional "Ciência, Tecnologia, Inovação" (POCTI)

Categories of matrices with only obvious Moore-Penrose inverses

AbstractLet R be an associative ring with 1 and with an involution a → ā, and let MR be the category of finite matrices over R with the involution (aij) → (aij)∗ = (āji). Then the following two statements are equivalent: (i) If A in MR has a Moore-Penrose inverse with respect to ∗, then A is permutationally equivalent to a matrix of the form B000 with B invertible. (ii) If 1 = ∑aā in R, then at most one of the a's is not zero

Drazin-Moore-Penrose invertibility in rings

Characterizations are given for elements in an arbitrary ring with involution, having a group inverse and a Moore-Penrose inverse that are equal and the difference between these elements and EP-elements is explained. The results are also generalized to elements for which a power has a Moore-Penrose inverse and a group inverse that are equal. As an application we consider the ring of square matrices of order $m$ over a projective free ring $R$ with involution such that $R^m$ is a module of finite length, providing a new characterization for range-Hermitian matrices over the complexes.Centro de Matemática da Universidade do Minho (CMAT).Fundação para a Ciência e a Tecnologia (FCT) - Programa Operacional "Ciência, Tecnologia, Inovação" (POCTI)

EP morphisms

AbstractThe 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 ∗ and with ∗-biproduct factorization, then a morphism of φ of C is EP iff there is a ∗-biproduct J ⊕ K and an invertible morphism θ : J → J such that φ is congruent to a morphism of the form θ 00 0: 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

Generalized inverses of morphisms with kernels

AbstractLet φ: X → Y be a morphism with kernel κ: K → X in an additive category with an involution ∗. Then φ has a Moore-Penrose inverse φ† with respect to ∗ iff φφ∗ + κ∗κ is invertible; in this case, φ† = φ∗ (φφ∗ + κ∗κ)−1. If X = Y, then φ has a group inverse φ# iff φ has a cokernel γ: X → K and φ2 + γκ is invertible; in this case, φ# = φ (φ2 + γκ)−1