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.
@article{bwmeta1.element.doi-10_1515_spma-2016-0010,
author = {Mika Mattila and Pentti Haukkanen},
title = {Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods},
journal = {Special Matrices},
volume = {4},
year = {2016},
pages = {101-109},
zbl = {1338.15076},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_spma-2016-0010}
}
Mika Mattila; Pentti Haukkanen. Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods. Special Matrices, Tome 4 (2016) pp. 101-109. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_spma-2016-0010/
[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. | Zbl 1144.15001
[2] M. Bahsi and S. Solak, Some particular matrices and their characteristic polynomials, Linear Multilinear Algebra 63 (2015) 2071–2078. [Crossref][WoS] | Zbl 1353.15029
[3] R. Bhatia, Infinitely divisible matrices, Amer. Math. Monthly 113 no. 3 (2006) 221–235. | Zbl 1132.15019
[4] R. Bhatia, Min matrices and mean matrices, Math. Intelligencer 33 no. 2 (2011) 22–28. [WoS] | Zbl 1247.15029
[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. | Zbl 1119.15012
[8] P. Haukkanen, On meet matrices on posets, Linear Algebra Appl. 249 (1996) 111–123. | Zbl 0870.15016
[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. | Zbl 1291.15049
[10] R. A. Horn and C. R. Johnson, Matrix Analysis, 1st ed., Cambridge University Press, 1985. | Zbl 0576.15001
[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] | Zbl 1102.62072
[12] I. Korkee, P. Haukkanen, On meet and join matrices associated with incidence functions, Linear Algebra Appl. 372 (2003) 127–153. [WoS] | Zbl 1036.06005
[13] I. Korkee and P. Haukkanen, On the divisibility of meet and join matrices, Linear Algebra Appl. 429 (2008) 1929–1943. [WoS] | Zbl 1157.11009
[14] M. Mattila and P. Haukkanen, Determinant and inverse of join matrices on two sets, Linear Algebra Appl. 438 (2013) 3891– 3904. | Zbl 1281.15036
[15] M. Mattila and P. Haukkanen, On the positive definiteness and eigenvalues of meet and join matrices, Discrete Math. 326 (2014) 9–19. [WoS] | Zbl 1290.15022
[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. | Zbl 1291.62014
[20] B.V. Rajarama Bhat, On greatest common divisor matrices and their applications, Linear Algebra Appl. 158 (1991) 77–97. | Zbl 0754.15012
[21] R. P. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth and Brooks/Cole, 1986. | Zbl 0608.05001