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

100%
Mattila M., Haukkanen P.
Special Matrices
|
2016
|
tom 4
|
nr 1
101-109
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.
2
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Bounds for sine and cosine via eigenvalue estimation

81%
Haukkanen P., Mattila M., Merikoski J. K., Kovacec A.
Special Matrices
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2014
|
tom 2
|
nr 1
EN
Define n × n tridiagonal matrices T and S as follows: All entries of the main diagonal of T are zero and those of the first super- and subdiagonal are one. The entries of the main diagonal of S are two except the (n, n) entry one, and those of the first super- and subdiagonal are minus one. Then, denoting by λ(·) the largest eigenvalue, [...] Using certain lower bounds for the largest eigenvalue, we provide lower bounds for these expressions and, further, lower bounds for sin x and cos x on certain intervals. Also upper bounds can be obtained in this way.
3
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On the spectral and Frobenius norm of a generalized Fibonacci r-circulant matrix

81%
Merikoski J. K., Haukkanen P., Mattila M., Tossavainen T.
Special Matrices
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2018
|
tom 6
|
nr 1
23-36
EN
Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g0, . . . , gn−1. We also give three lower bounds (with equality conditions) for the spectral norm of this matrix. For this purpose, we present three ways to estimate the spectral norm from below in general.
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