Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access January 21, 2016

Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods

  • Mika Mattila and Pentti Haukkanen
From the journal Special Matrices


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.


[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. 10.1155/2007/37580Search in Google Scholar

[2] M. Bahsi and S. Solak, Some particular matrices and their characteristic polynomials, Linear Multilinear Algebra 63 (2015) 2071–2078. 10.1080/03081087.2014.940940Search in Google Scholar

[3] R. Bhatia, Infinitely divisible matrices, Amer. Math. Monthly 113 no. 3 (2006) 221–235. 10.1080/00029890.2006.11920300Search in Google Scholar

[4] R. Bhatia, Min matrices and mean matrices, Math. Intelligencer 33 no. 2 (2011) 22–28. 10.1007/s00283-010-9194-zSearch in Google Scholar

[5] K. L. Chu, S. Puntanen and G. P. H. Styan, Problem section, Stat Papers 52 (2011) 257–262. 10.1007/s00362-010-0363-0Search in Google Scholar

[6] R. Davidson and J. G. MacKinnon, Econometric Theory and Methods, Oxford University Press, 2004. Search in Google Scholar

[7] C. M. da Fonseca, On the eigenvalues of some tridiagonal matrices, J. Comput. Appl. Math. 200 no. 1 (2007) 283–286. 10.1016/ in Google Scholar

[8] P. Haukkanen, On meet matrices on posets, Linear Algebra Appl. 249 (1996) 111–123. 10.1016/0024-3795(95)00349-5Search in Google Scholar

[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. Search in Google Scholar

[10] R. A. Horn and C. R. Johnson, Matrix Analysis, 1st ed., Cambridge University Press, 1985. 10.1017/CBO9780511810817Search in Google Scholar

[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. 10.1080/03610920600672146Search in Google Scholar

[12] I. Korkee, P. Haukkanen, On meet and join matrices associated with incidence functions, Linear Algebra Appl. 372 (2003) 127–153. 10.1016/S0024-3795(03)00497-XSearch in Google Scholar

[13] I. Korkee and P. Haukkanen, On the divisibility of meet and join matrices, Linear Algebra Appl. 429 (2008) 1929–1943. 10.1016/j.laa.2008.05.025Search in Google Scholar

[14] M. Mattila and P. Haukkanen, Determinant and inverse of join matrices on two sets, Linear Algebra Appl. 438 (2013) 3891– 3904. 10.1016/j.laa.2011.12.014Search in Google Scholar

[15] M. Mattila and P. Haukkanen, On the positive definiteness and eigenvalues of meet and join matrices, Discrete Math. 326 (2014) 9–19. 10.1016/j.disc.2014.02.018Search in Google Scholar

[16] L. A. Moyé, Statistical Monitoring of Clinical Trials, 1st ed., Springer, 2006. Search in Google Scholar

[17] H. Neudecker, G. Trenkler, and S. Liu, Problem section, Stat Papers 50 (2009) 221–223. 10.1007/s00362-008-0174-8Search in Google Scholar

[18] G. Pólya and G. Szegö, Problems and Theorems in Analysis II, Vol. II, 4th ed., Springer, 1971. 10.1007/978-1-4757-1640-5Search in Google Scholar

[19] S. Puntanen, G. P. H. Styan, and J. Isotalo, Matrix Tricks for Linear Statistical Models -Our Personal Top Twenty, 1st ed., Springer, 2011. 10.1007/978-3-642-10473-2_1Search in Google Scholar

[20] B.V. Rajarama Bhat, On greatest common divisor matrices and their applications, Linear Algebra Appl. 158 (1991) 77–97. 10.1016/0024-3795(91)90051-WSearch in Google Scholar

[21] R. P. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth and Brooks/Cole, 1986. 10.1007/978-1-4615-9763-6_1Search in Google Scholar

Received: 2015-9-25
Accepted: 2016-1-2
Published Online: 2016-1-21

©2016 Mika Mattila and Pentti Haukkanen

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 11.12.2023 from
Scroll to top button