Robert E. Hartwig

Some additive results on Drazin inverse

Some additive perturbation results for Drazin inverses are given. In particular, a formula is given for the Drazin inverse of a sum of two matrices, when one of the products of these matrices vanishes. Some special applications of this are also considered.

Block generalized inverses

The existence of the Moore-Penrose inverse is discussed for elements of a *-regular ring R. A technique is developed for computing conditional and reflexive inverses for matrices in R2×2, which is then used to calculate the Moore-Penrose inverse for these matrices. Several applications are given, generalizing many of the classical results; in particular, we shall emphasize the cases of bordered matrices, Schur complements, block-rank formulae and EP elements.

GROUP INVERSES AND DRAZIN INVERSES OF BIDIAGONAL AND TRIANGULAR TOEPLITZ MATRICES

Necessary and sufficient conditions are given for the existence of the group and Drazin inverses of bidiagonal and triangular Toeplitz matrices over an arbitrary ring.

Matrices for which A∗ and A† commute

The class of complex star-dagger matrices for which A∗ and A† commute, is investigated. Some fundamental properties are given and a canonical form is derived. The interaction of this class of matrices with other common classes of special matrices, which as group matrices, EP matrices, and idempotents, will be looked into

On some characterizations of the “star” partial ordering for matrices and rank subtractivity

Abstract The main result is that Drazins “star” partial ordering A ⩽ ∗ B holds if and only if A ∠ B and B † −A † =(B−A) † , where A ⩽ ∗ B is defined by A ∗ A = A ∗ B and AA ∗ = BA ∗ , and where A ∠ B denotes rank subtractivity. Here A and B are m × n complex matrices and the superscript † denotes the Moore-Penrose inverse. Several other characterizations of A ⩽ ∗ B are given, with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger “star” order; a key tool is a new canonical form for rank subtractivity. Connections with simultaneous singular-value decompositions, Schur complements, and idempotent matrices are also mentioned.

The reverse order law revisited

Abstract Necessary and sufficient conditions are given for the triple reverse order law (ABC)† = C†B†A† to hold. Some special cases are considered.

The matrix equation A X – XB = C and its special cases

The consistency and solutions of the matrix equations

Asymptotic behavior of Toeplitz matrices and determinants

A\bar X - XB = C

The group inverse of a companion matrix

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Partial Orders Based on Outer Inverses

A\bar X \pm XA^T = C