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2016 | 4 | 1 | 101-109
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Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let T = {z1, z2, . . . , zn} be a finite multiset of real numbers, where z1 ≤ z2 ≤ · · · ≤ zn. The purpose of this article is to study the different properties of MIN and MAX matrices of the set T with min(zi , zj) and max(zi , zj) as their ij entries, respectively.We are going to do this by interpreting these matrices as so-called meet and join matrices and by applying some known results for meet and join matrices. Once the theorems are found with the aid of advanced methods, we also consider whether it would be possible to prove these same results by using elementary matrix methods only. In many cases the answer is positive.
Słowa kluczowe
Wydawca
Czasopismo
Rocznik
Tom
4
Numer
1
Strony
101-109
Opis fizyczny
Daty
otrzymano
2015-09-25
zaakceptowano
2016-01-02
online
2016-01-21
Twórcy
autor
  • Department of Mathematics, Tampere University of Technology, Finland
  • School of Information Sciences, University of Tampere, Finland
Bibliografia
  • [1] E. Altinisik, N. Tuglu, and P. Haukkanen, Determinant and inverse of meet and join matrices, Int. J. Math. Math. Sci. 2007 (2007) Article ID 37580.
  • [2] M. Bahsi and S. Solak, Some particular matrices and their characteristic polynomials, Linear Multilinear Algebra 63 (2015) 2071–2078. [Crossref][WoS]
  • [3] R. Bhatia, Infinitely divisible matrices, Amer. Math. Monthly 113 no. 3 (2006) 221–235.
  • [4] R. Bhatia, Min matrices and mean matrices, Math. Intelligencer 33 no. 2 (2011) 22–28. [WoS]
  • [5] K. L. Chu, S. Puntanen and G. P. H. Styan, Problem section, Stat Papers 52 (2011) 257–262.
  • [6] R. Davidson and J. G. MacKinnon, Econometric Theory and Methods, Oxford University Press, 2004.
  • [7] C. M. da Fonseca, On the eigenvalues of some tridiagonal matrices, J. Comput. Appl. Math. 200 no. 1 (2007) 283–286.
  • [8] P. Haukkanen, On meet matrices on posets, Linear Algebra Appl. 249 (1996) 111–123.
  • [9] P. Haukkanen, M. Mattila, J. K. Merikoski, and A. Kovačec, Bounds for sine and cosine via eigenvalue estimation, Spec. Matrices 2 no. 1 (2014) 19–29.
  • [10] R. A. Horn and C. R. Johnson, Matrix Analysis, 1st ed., Cambridge University Press, 1985.
  • [11] J. Isotalo and S. Puntanen, Linear prediction suflciency for new observations in the general Gauss–Markov model, Comm. Statist. Theory Methods 35 (2006) 1011–1023. [Crossref]
  • [12] I. Korkee, P. Haukkanen, On meet and join matrices associated with incidence functions, Linear Algebra Appl. 372 (2003) 127–153. [WoS]
  • [13] I. Korkee and P. Haukkanen, On the divisibility of meet and join matrices, Linear Algebra Appl. 429 (2008) 1929–1943. [WoS]
  • [14] M. Mattila and P. Haukkanen, Determinant and inverse of join matrices on two sets, Linear Algebra Appl. 438 (2013) 3891– 3904.
  • [15] M. Mattila and P. Haukkanen, On the positive definiteness and eigenvalues of meet and join matrices, Discrete Math. 326 (2014) 9–19. [WoS]
  • [16] L. A. Moyé, Statistical Monitoring of Clinical Trials, 1st ed., Springer, 2006.
  • [17] H. Neudecker, G. Trenkler, and S. Liu, Problem section, Stat Papers 50 (2009) 221–223.
  • [18] G. Pólya and G. Szegö, Problems and Theorems in Analysis II, Vol. II, 4th ed., Springer, 1971.
  • [19] S. Puntanen, G. P. H. Styan, and J. Isotalo, Matrix Tricks for Linear Statistical Models -Our Personal Top Twenty, 1st ed., Springer, 2011.
  • [20] B.V. Rajarama Bhat, On greatest common divisor matrices and their applications, Linear Algebra Appl. 158 (1991) 77–97.
  • [21] R. P. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth and Brooks/Cole, 1986.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_spma-2016-0010
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