Roland Puystjens

The group inverse of a companion matrix

A complete characterization is given for the group inverse of a companion matrix over an arbitrary ring to exist. Formulae are given for the actual group inverse and some consequences are drawn.

The Moore-Penrose inverse of a morphism with factorization

Abstract Given an m -by- n matrix A of rank r over a field with an involutory automorphism, it is well known that A has a Moore-Penrose inverse if and only if rank A ∗ A = r = rank AA ∗ . By use of the full-rank factorization theorem, this result may be restated in the category of finite matrices as follows: if ( A 1 , r , A 2 ) is an (epic, monic) factorization of A : m → n through r , then A has a Moore-Penrose inverse if and only if ( A ∗ A 1 , r , A 2 ) and ( A 1 , r , A 2 A ∗ ) are, respectively, (epic, monic) factorizations of A ∗ A : n → n and AA ∗ : m → m through r . This characterization of the existence of Moore-Penrose inverses is extended to arbitrary morphisms with (epic, monic) factorizations.

About the von Neumann regularity of triangular block matrices.

Abstract 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.

Drazin invertibility for matrices over an arbitrary ring

Abstract Characterizations are given for existence of the Drazin inverse of a matrix over an arbitrary ring. Moreover, the Drazin inverse of a product PAQ for which there exist a P′ and Q′ such that P′PA=A=AQQ′ can be characterized and computed. This generalizes recent results obtained for the group inverse of such products. The results also apply to morphisms in (additive) categories. As an application we characterize Drazin invertibility of companion matrices over general rings.

The moore-penrose inverse of a matrix over a semi-simpie artinian ring

Given an m-by-n matrix over a semi-simple artinian ring with aninvolutory automorphism, necessary and sufficient conditions are given for the matrix tao have a Moore—Penrose inverse. Moreover, expressions for the MP-inverse are obtained. These formulas may be considered as a generalization of the MacDuffee formula for the MP-inverse of a matrix over the complex numbers.

Symmetric morphisms and the existence of Moore-Penrose inverses

Abstract Let φ: X → Y be a morphism of an additive category with an involution ∗ . Then φ has a Moore-Penrose inverse with respect to ∗ iff φφ ∗ +η ∗ η is invertible for some annihilator η: N → X of φ iff there exists an object Z and morphisms α: X → Z , β: Z → Z , γ: Z → Y such that φ = αβγ , β 2 = β = β ∗ , αβ = α , βγ = γ , and α ∗ α+ 1 z – β and γγ ∗ +1 z – β are invertible. In this case, φ † = φ ∗ (φφ ∗ +η ∗ η) −1 = γ ∗ (γγ ∗ +1 z – β) -1 (α ∗ α+1 z – β) -1 α ∗ .

Generalized inverses of morphisms with kernels

Abstract Let 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 φ 2 + γκ is invertible; in this case, φ # = φ ( φ 2 + γκ ) −1 .

Categories of matrices with only obvious Moore-Penrose inverses

Abstract Let 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 B 0 0 0 with B invertible. (ii) If 1 = ∑aā in R, then at most one of the as is not zero.