pedro patricio
departamento de matematica
universidade do minho
4710-057 braga, portugal
email: pedro(AT)math(DOT)uminho(DOT)pt
tel: +351.
fax: +351.

[ scientific interests/teaching ] [ publications ] [ research colaboration ] [ math stuff ] [ non-math stuff ] [ linux links ] [ blog ]


my scientic interests are in the area of generalized inverses, thru an algebraic approach. this includes g.i. of matrices over rings, partial orderings, g.i. on rings and semigroups...
i did my PhD under supervision of Roland Puystjens, from the RUG, Gent, Belgium. until he retired, we continued to do some joint work. part of this work was jointly with R.E. Hartwig, from NCSU, USA, with whom i have been working. more recently, two colleagues from my department started to be interested on generalized inverses, and eventually we submitted and got our papers accepted. one of these results led to recent research colaboration with nieves castro-gonzalez. in the meanwhile, among the several courses i taught, some caught my real interest either by the way they were taught, or by the course itself. these, occasionally, led to some course notes. other math interests include algebraic coding theory (block codes and convolutional codes) and cryptography.

SOME courses i taught RECENTLY (and enjoyed)
  1. teoria de codigos - coding theory (with mupad) (2005/06)
  2. matematica discreta - discrete mathematics (2005/06)
  3. algebra linear (para engenheiros) - linear algebra (with octave) (2008/09)
  4. teoria de numeros computacional - computational number theory (with pari/gp) (2006/07, 2008/09)

Make the Move!
From Windows to Linux

list of publications
  1. Patricio, P., The Moore-Penrose inverse of von Neumann regular matrices over a ring, Linear Algebra and Its Applications, 332-334:469-483, 2001.
  2. Patricio, P. and Puystjens, R., About the von Neumann regularity of triangular block matrices, Linear Algebra and Its Applications, 332-334:485-502, 2001.
  3. Koliha, J.J. and Patricio, P., Elements of rings with equal spectral idempotents, Journal of the Australian Mathematical Society, 72:137-152, 2002.
  4. Patricio, P., The Moore-Penrose inverse of a factorization, Linear Algebra and Its Applications, 370:227-235, 2003.
  5. P. Patricio and R. Puystjens, Generalized invertibility in two semigroups of a ring, Linear Algebra and Its Applications, 377:125-139, 2004.
  6. P. Patricio and R. Puystjens, Drazin-Moore-Penrose invertibility in rings, Linear Algebra and Its Applications, 389:159-173, 2004.
  7. P. Patricio, R.E. Hartwig and R. Puystjens, Diagonalizing matrices via orthogonal Pierce decompositions, Linear Algebra and Its Applications, 381-391, 2005.
  8. P. Patricio and R.E. Hartwig, On Roth's pseudo equivalence over rings, ELA vol 16:111-124, 2007.
  9. Patricio, Pedro, Veloso da Costa, A., On the Drazin index of regular elements, Central European Journal of Mathematics, 7(2):200--205, 2009.
  10. Patricio, Pedro, Hartwig, R.E., Some additive results on Drazin inverses, Applied Mathematics and Computation, 215 (2009), 530-538.
  11. Patricio, Pedro, Mendes Araujo, C., Moore-Penrose invertibility in involutory rings: the case aa+=bb+, Linear and Multilinear Algebra 58(4) (2010), 445-452
  12. Castro-Gonzalez, N., Mendes Araujo, C., Patricio, P., "Generalized inverses of a sum in rings", Bulletin of the Australian Mathematical Society, 82 (2010), 156-164
  13. Patricio, P., Hartwig, R.E., "The (2,2,0) Group Inverse Problem", Applied Mathematics and Computation 217(2) (2010), 516-520.
  14. Patricio, P., Hartwig, R.E., "The link between regularity and strong-pi-regularity", Journal of the Australian Mathematical Society 89 (2010), 17-22.
  15. Hartwig, R.E., Patricio, P., "When does the Moore-Penrose inverse flip?", Operators and Matrices, 6 (1) (2012), 181-192.
  16. Patricio, P., "The Moore-Penrose inverse of a companion matrix", Linear Algebra and Its Applications 437(3) (2012), 870-877.
  17. Hartwig, R.E., Patricio, P., "A note on power bounded matrices", The Electronic Journal of Linear Algebra 23 (2012), 625-645.
  18. Mary, X., Patricio, P., "The inverse along a lower triangular matrix", Applied Mathematics and Computation 219(3) (2012), 886-891
  19. Patricio, P., Hartwig, R.E.; "The (2,2,0) Drazin inverse problem", Linear Algebra and its Applications 437(11) (2012), 2755-2772
  20. Lebtahi, L., Patricio, P., Thome, N., "Special elements in a ring related to Drazin inverses", Linear and Multilinear Algebra 61(8) (2013), 1017-1027.
  21. Mary, X., Patricio, P., "Generalized inverses modulo H in semigroups and rings", Linear and Multilinear Algebra 61(8) (2013), 1130-1135.
  22. Lebtahi, L., Patricio, P., Thome, N., "The diamond partial order in rings", Linear and Multilinear Algebra 62(3) (2014), 386-395.
  23. Hartwig, R.E., Patricio, P., "Two by two units", Electron. J. Linear Algebra 27 (2014), 489-503.
  24. Zhu, Huihui; Chen, Jianlong; Zhang, Xiaoxiang; Patricio, Pedro, "The Moore-Penrose inverse of 2x2 matrices over a certain *-regular ring". Appl. Math. Comput. 246 (2014), 263-267.
  25. Patricio, P., Hartwig, R.E., "Some regular sums", Linear and Multilinear Algebra, 63(1) (2015), 185-200.
  26. Zhu, Huihui; Chen, Jianlong; Patricio, Pedro, "Representations for the pseudo Drazin inverse of elements in a Banach algebra", Taiwanese J. Math. 19 (2015), no. 2, 349-362.
  27. Burgos, M.; Marquez-Garcia, M.C.; Patricio, P.; "On mappings preserving the sharp and star orders", Linear Algebra and Its Applications, 483 (2015), 268-292.
  28. Zhu, Huihui; Jianlong; Patricio, Pedro; Zhang, Yulin, "The inverse along a product and its applications", Linear Multilinear Algebra 64, No. 5, 834-841 (2016).
  29. Zhu, Huihui; Chen, Jianlong; Patricio, Pedro, "Further results on the inverse along an element in semigroups and rings", Linear Multilinear Algebra 64, No. 3, 393-403 (2016).
  30. Mary, Xavier; Patricio, Pedro, "The group inverse of a product", Linear and Multilinear Algebra 64, No 9, 1776-1784 (2016).
  31. H.H. Zhu, J.L. Chen, P. Patricio, "The Moore-Penrose inverse of differences and products of projectors in a ring with involution", Turk J Math, 40, (2016), 1316-1324.
  32. H.H. Zhu, J.L. Chen, P. Patricio, "Reverse order law for the inverse along an element", Linear Multilinear Algebra 65, 166-177 (2017).
  33. J.L. Chen, H.H. Zhu, P. Patricio, Y.L. Zhang, "Characterizations and representations of core and dual core inverses", Canadian Mathematical Bulletin 60(2017), no. 2, 269-282.
  34. H. Zou, J.L. Chen, P. Pedro Patricio, "Characterizations of m-EP elements in rings", Linear Multilinear Algebra, to appear.
  35. H.H. Zhu, J.L. Chen, P. Patricio, X. Mary, "Centralizer's applications to the inverse along an element", Applied Mathematics and Computation, to appear.

research colaboration:

member of the centro de matematica - cmat, SPM and ILAS

math stuff:

my Erdos numer is 4.
the Erdos number of a person is the length of a shortest chain of joint mathematical publications leading to the famous mathematician Paul Erdos (see all about that here). My chain:

(Erdos, Rodl),   (Rodl, Muller),   (Muller, Koliha),   (Koliha,Patricio)

P. Erdos and V. Rodl, Coverings of r-graphs by complete r-partite subgraphs, Random Structures Algorithms 6 (1995), 2-3.

V. Muller and V. Rodl, Monotone paths in ordered graphs, Combinatorica 2 (1982), 193-201.

J. J. Koliha, M. Mbekhta, V. Muller and P. W. Poon, Corigendum and addendum: "On the axiomatic theory of the spectrum II", Studia Mathematica 130 (1998), 193-198.

another path (from the mathscinet):
Pedro Patricio coauthored with Robert E. Hartwig MR2134973 (2006b:15007)
Robert E. Hartwig coauthored with Frank J. Hall MR0404288 (53 #8090)
Frank J. Hall coauthored with Guantao Chen MR1821633 (2001m:05121)
Guantao Chen coauthored with Paul Erdos MR1215330 (94a:05148)


Bulletin of the Belgium Math. Soc. - Simon Stevin

Central European Journal of Mathematics

Journal of the Australian Mathematical Society

Linear Algebra and Its Applications

Mathematica Sinica

Soochow Journal of Mathematics

non-math stuff:

i am interested by everything concerning hi-fi, but i am far from being an audiophile. in any case, i have as an universal digital source a denon 2900 (super-audio CD, DVD-audio) and a denon 3803 as a receiver.
visonik was the choice for loudspeakers: two pairs of evolution 6 for front+surround and one evolution center as central channel. interconnects include nordost, qed and profigold.
an important source of reviews and news is the home cinema choice site, with lots of tests and comparisons. discussion foruns i try to follow include dvdmania, forum hi-fi, avforum, avforums. PT source of news and reviews: hificlube

NOTICE: this is quite old; i'll update it whenever i find the time and mood :-)
i have been using linux since 2000. i have used mandrake (now mandriva) and suse (which i really recommed, namely the 10.0). at home i have and old compaq armada E700 laptop (more like a brick, nowadays), with a pentium II @ astonishing ;-) 400MHz, together with 128Mb ram and 20G hard drive. it drags quite a bit, and some patience is required. it is running mepis, a former debian based distro, and since 2006 a ubuntu based distro (screenshot here). at my office, i was fortunate enough to receive a 3.0GHz dual core pentium, with 512Mb ram. after using mepis, i experienced and was delighted to use kororaa, a gentoo based distro, with XGL support (see screenshots here and here). i have, in october '06, upgraded to Kororaa AIGLXgl (v 0.3), running on AIXGL (screenshot here and here). in the labs i use fedora-core, and centOS in our server (where this page is).
preferred software: the firefox family, kile+latex, octave/pari-gp/mupad, emacs, vi ;-) and kwrite, gimp (!), openoffice for received (M$)office documents.
info on this subject can be found at /., and it's portuguese version gildot. i also check for news and discussions in the spanish barrapunto and in the brazilian vivaolinux. i keep track on the distro's update by checking (RSS feeds!!!), and