@article{hartwig_patricio_2023, title={Divisibility of Finite Geometric Series}, volume={46}, ISSN={["2180-4206"]}, DOI={10.1007/s40840-023-01556-5}, abstractNote={Abstract}, number={5}, journal={BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY}, author={Hartwig, Robert E. and Patricio, Pedro}, year={2023}, month={Sep} }
 @article{hartwig_patricio_2023, title={The GCD for finite geometric series}, volume={37}, ISSN={["0354-5180"]}, DOI={10.2298/FIL2329973H}, number={29}, journal={FILOMAT}, author={Hartwig, R. E. and Patricio, Pedro}, year={2023}, pages={9973–9979} }
 @article{patricio_hartwig_2021, title={From Euclid to Corner Sums - a Trail of Telescoping Tricks}, volume={35}, ISSN={["0354-5180"]}, DOI={10.2298/FIL2114613P}, abstractNote={Euclid?s algorithm is extended to binomials, geometric sums and corner sums.
   Two-sided non-commuting, non-constant linear difference equations will be
   solved, and the solution is applied to corner sums, thereby presenting an
   explicit formula for the generator of the bi-module spanned by the two
   starting corner sums.}, number={14}, journal={FILOMAT}, author={Patricio, Pedro and Hartwig, Robert E.}, year={2021}, pages={4613–4636} }
 @article{castro-gonzalez_hartwig_2018, title={PERTURBATION RESULTS AND THE FORWARD ORDER LAW FOR THE MOORE-PENROSE INVERSE OF A PRODUCT}, volume={34}, DOI={10.13001/1081-3810,1537-9582.3365}, journal={ELECTRONIC JOURNAL OF LINEAR ALGEBRA}, author={Castro-Gonzalez, Nieves and Hartwig, Robert E.}, year={2018}, pages={514–525} }
 @article{hartwig_patricio_2018, title={The group inverse of the nivellateur}, volume={66}, ISSN={["1563-5139"]}, DOI={10.1080/03081087.2017.1397092}, abstractNote={ABSTRACT We shall derive necessary and sufficient conditions for the nivellateur to have a group inverse over an algebraically closed field. We then extend these results to arbitrary fields.}, number={12}, journal={LINEAR & MULTILINEAR ALGEBRA}, author={Hartwig, R. E. and Patricio, P.}, year={2018}, pages={2409–2420} }
 @article{patricio_hartwig_2015, title={Some regular sums}, volume={63}, ISSN={["1563-5139"]}, DOI={10.1080/03081087.2013.860592}, abstractNote={In this paper, we examine the question of regularity of sums of special elements that appear in the study of orthogonality and invertibility.}, number={1}, journal={LINEAR & MULTILINEAR ALGEBRA}, author={Patricio, P. and Hartwig, R. E.}, year={2015}, month={Jan}, pages={185–200} }
 @article{hartwig_patricio_2014, title={Two by two units}, volume={27}, DOI={10.13001/1081-3810.1632}, abstractNote={In this paper, we will use outer inverses and the Brown-McCoy shift to characterize the existence of the inverse and group inverse of a 2 × 2 block matrix.}, journal={Electronic Journal of Linear Algebra}, author={Hartwig, R. E. and Patricio, P.}, year={2014}, pages={489–503} }
 @article{hartwig_patricio_2012, title={A note on power bounded matrices}, volume={23}, DOI={10.13001/1081-3810.1546}, abstractNote={This research was financed by FEDER Funds through ``Programa Operacional Factores de Competitividade - COMPETE}, journal={Electronic Journal of Linear Algebra}, author={Hartwig, R. E. and Patricio, P.}, year={2012}, pages={625–645} }
 @article{patricio_hartwig_2010, title={THE LINK BETWEEN REGULARITY AND STRONG-PI-REGULARITY}, volume={89}, ISSN={["1446-8107"]}, DOI={10.1017/s1446788710001448}, abstractNote={Abstract}, number={1}, journal={JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY}, author={Patricio, Pedro and Hartwig, R. E.}, year={2010}, month={Aug}, pages={17–22} }
 @article{patricio_hartwig_2010, title={The (2,2,0) group inverse problem}, volume={217}, ISSN={["0096-3003"]}, DOI={10.1016/j.amc.2010.05.084}, abstractNote={We characterize the existence of the group inverse of a two by two matrix with zero (2, 2) entry, over a ring by means of the existence of the inverse of a suitable function of the other three entries. Some special cases are derived.}, number={2}, journal={APPLIED MATHEMATICS AND COMPUTATION}, author={Patricio, P. and Hartwig, R. E.}, year={2010}, month={Sep}, pages={516–520} }
 @article{patricio_hartwig_2009, title={Some additive results on Drazin inverses}, volume={215}, DOI={10.1016/j.amc.2009.05.021}, abstractNote={In this paper, some additive results on Drazin inverse of a sum of Drazin invertible elements are derived. Some converse results are also presented.}, number={2}, journal={Applied Mathematics and Computation}, author={Patricio, P. and Hartwig, R. E.}, year={2009}, pages={530–538} }
 @article{hartwig_patricio_2007, title={On Roth's pseudo equivalence over rings}, volume={16}, DOI={10.13001/1081-3810.1187}, abstractNote={The pseudo-equivalence of a block lower triangular matrix T =( Tij )o ver ar egular ring and its block diagonal matrix D(T )=( Tii) is characterized in terms of suitable Roth consistency conditions. The latter can in turn be expressed in terms of the solvability of certain matrix equations of the form TiiXYT jj = Uij.}, journal={Electronic Journal of Linear Algebra}, author={Hartwig, R. E. and Patricio, P.}, year={2007}, pages={111–124} }
 @article{hartwig_2006, title={Note on a linear difference equation}, volume={113}, ISSN={["0002-9890"]}, DOI={10.2307/27641892}, number={3}, journal={AMERICAN MATHEMATICAL MONTHLY}, author={Hartwig, RE}, year={2006}, month={Mar}, pages={250–256} }
 @article{chen_hartwig_2006, title={The Picard iteration and its application}, volume={54}, ISSN={["0308-1087"]}, DOI={10.1080/03081080500209703}, abstractNote={The convergence behavior of the Picard iteration Xk+1=AXk+B and the weighted case Yk=Xk/bk is investigated. It is shown that the convergence of both these iterations is related to the so-called effective spectrum of A with respect to some matrix. As an application of our convergence results we discuss the convergence behavior of a sequence of scaled triangular matrices {DNTN }.}, number={5}, journal={LINEAR & MULTILINEAR ALGEBRA}, author={Chen, X and Hartwig, RE}, year={2006}, month={Sep}, pages={329–341} }
 @article{hartwig_patricio_puystjens_2005, title={Diagonalizing triangular matrices via orthogonal Pierce decompositions}, volume={401}, ISSN={["1873-1856"]}, DOI={10.1016/j.laa.2004.10.002}, abstractNote={A class of sufficient conditions is given to ensure that the sum a + b in a ring R, is equivalent to a sum x + y, which is an orthogonal Pierce decomposition.This is then used to show that a lower triangular matrix, with a regular diagonal is equivalent to its diagonal iff the matrix admits a lower triangular von Neumann inverse.}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Hartwig, RE and Patricio, P and Puystjens, R}, year={2005}, month={May}, pages={381–391} }
 @article{hartwig_maroulas_2005, title={Focal power}, volume={13}, DOI={10.13001/1081-3810.1150}, abstractNote={The properties of a 2 × 2 block matrix M over a field for which (M)11 = (M)11 are examined. This “fire-wall” property will be characterized by the vanishing of the sequence of moments Fi = CDiB.}, journal={Electronic Journal of Linear Algebra}, author={Hartwig, R. E. and Maroulas, J.}, year={2005}, pages={56–71} }
 @article{hartwig_li_wei_2005, title={Representations for the Drazin inverse of a 2 x 2 block matrix}, volume={27}, ISSN={["1095-7162"]}, DOI={10.1137/040606685}, abstractNote={Two representations for the Drazin inverse of a $2\times2$ block matrix $M=[{A \atop C}\;{B \atop D}]$, where $A$ and $D$ are square matrices, in terms of the Drazin inverses of $A$ and $D$ have been recently developed under the assumptions that $C(I-AA^{D})=0$ and $(I-AA^{D})B=0$, and that the generalized Schur complement $D-CA^{D}B$ is either nonsingular or zero. These two representations of $M^{D}$ are extended to the case where $C(I-AA^{D})=0$ and $(I-AA^{D})B=0$ are substituted with $C(I-AA^{D})B=0$ and $A(I-AA^{D})B=0$. Moreover, upper bounds for the index of $M$ are studied. Numerical examples are given to illustrate the new results.}, number={3}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Hartwig, R and Li, XZ and Wei, YM}, year={2005}, pages={757–771} }
 @article{chen_hartwig_2005, title={The semi-iterative method applied to the hyper-power iteration}, volume={12}, ISSN={["1070-5325"]}, DOI={10.1002/nla.429}, abstractNote={The semi‐iterative method (SIM) is applied to the hyper‐power (HP) iteration, and necessary and sufficient conditions are given for the convergence of the semi‐iterative–hyper‐power (SIM–HP) iteration. The root convergence rate is computed for both the HP and SIM–HP methods, and the quotient convergence rate is given for the HP iteration. Copyright © 2005 John Wiley & Sons, Ltd.}, number={9}, journal={NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS}, author={Chen, XZ and Hartwig, RE}, year={2005}, month={Nov}, pages={895–910} }
 @article{poplin_hartwig_2004, title={Determinantal identities over commutative semirings}, volume={387}, ISSN={["0024-3795"]}, DOI={10.1016/j.laa.2004.02.019}, abstractNote={We present a development of determinantal identities over commutative semirings. This includes a generalization of the Cauchy–Binet and Laplace Theorems, as well as results on compound matrices and adjoints. It is further shown that Laplace's Theorem is a special case of the grade-s-adjoint identity.}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Poplin, PL and Hartwig, RE}, year={2004}, month={Aug}, pages={99–132} }
 @article{hartwig_wang_wei_2001, title={Some additive results on Drazin inverse}, volume={322}, ISSN={["1873-1856"]}, DOI={10.1016/S0024-3795(00)00257-3}, abstractNote={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.}, number={1-3}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Hartwig, RE and Wang, GR and Wei, YM}, year={2001}, month={Jan}, pages={207–217} }
 @article{hartwig_prasad_2000, title={Creation and annihilation in matrix theory}, volume={305}, ISSN={["0024-3795"]}, DOI={10.1016/s0024-3795(99)00197-4}, abstractNote={A closed form representation is given for the matrix S that sweeps out a single column. This includes the integer as well as the unitary and regular cases. The closed form is built up from the 2×2 case. The product rule for adjoints is used to show that the dual of this procedure is precisely the completion procedure, which completes a single column to a matrix B such that SB=BS=diagonal. This construction underlines the fact that the completion and elimination processes are complementary. By permuting rows suitably, the matrices may be assumed to be in lower Hessenberg form.}, number={1-3}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Hartwig, RE and Prasad, KM}, year={2000}, month={Jan}, pages={47–65} }
 @article{hartwig_semrl_1999, title={Constrained convergence}, volume={29}, ISSN={["0035-7596"]}, DOI={10.1216/rmjm/1181071685}, number={1}, journal={ROCKY MOUNTAIN JOURNAL OF MATHEMATICS}, author={Hartwig, RE and Semrl, P}, year={1999}, pages={177–195} }
 @article{hartwig_semrl_1998, title={Power additivity and orthogonality}, volume={20}, ISSN={["0895-4798"]}, DOI={10.1137/S089547989731498X}, abstractNote={The relation between rank additivity and orthogonality is analyzed. The central role played by Cochran's theorem is illustrated and the properties of orthogonal and power-additive families are investigated.}, number={1}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Hartwig, RE and Semrl, P}, year={1998}, month={Sep}, pages={1–13} }
 @article{mani_hartwig_1997, title={Generalized polynomial bases and the Bezoutian}, volume={251}, ISSN={["0024-3795"]}, DOI={10.1016/0024-3795(95)00585-4}, abstractNote={A foundation polynomial is used to induce polynomial bases for Fn−1[x], the vector space of polynomials of degree less than n over an arbitrary field F. The associated bases are then used to block diagonalize the Bezout matrix of two polynomials under congruence.}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Mani, V and Hartwig, RE}, year={1997}, month={Jan}, pages={293–320} }
 @article{hartwig_katz_1997, title={On products of EP matrices}, volume={252}, ISSN={["0024-3795"]}, DOI={10.1016/0024-3795(95)00693-1}, abstractNote={Necessary and sufficient conditions are given for the product of two EP matrices to be EP. Some special cases are considered.}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Hartwig, RE and Katz, IJ}, year={1997}, month={Feb}, pages={339–345} }
 @article{chen_hartwig_1997, title={On the convergence of power scaled Cesaro sums}, volume={267}, ISSN={["0024-3795"]}, DOI={10.1016/S0024-3795(97)80056-0}, abstractNote={Power scaling is combined with Cesáro summation, and necessary and sufficient local conditions are given for the resulting matrix sequence JN(T) to converge. It is shown that this convergence is closely related to path conditions on the one hand as well as suitable bilinear consistency conditions on the other hand.}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Chen, XZ and Hartwig, RE}, year={1997}, month={Dec}, pages={335–358} }