Determinants of some pentadiagonal matrices

In this paper we consider pentadiagonal \((n+1)\times(n+1)\) matrices with two subdiagonals and two superdiagonals at distances \(k\) and \(2k\) from the main diagonal where \(1\le k \lt 2k\le n\). We give an explicit formula for their determinants and also consider the Toeplitz and “imperfect” Toeplitz versions of such matrices. Imperfectness means that the first and last \(k\) elements of the main diagonal differ from the elements in the middle. Using the rearrangement due to Egerváry and Szász we also show how these determinants can be factorized.

On some reciprocal matrices with elliptical components of their Kippenhahn curves

By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai , i +1 ai +1, i = 1 for i = 1, . . ., n − 1. We establish some properties of the numerical range generating curves C(A) (also called Kippenhahn curves) of such matrices, in particular concerning the location of their elliptical components. For n ≤ 6, in particular, we describe completely the cases when C(A) consist entirely of ellipses. As a corollary, we also provide a complete description of higher rank numerical ranges when these criteria are met.

Measure of Departure from Conditional Symmetry Based on Cumulative Probabilities for Square Contingency Tables

For the analysis of square contingency tables with ordered categories, a measure was developed to represent the degree of departure from the conditional symmetry model in which there is an asymmetric structure of the cell probabilities with respect to the main diagonal of the table. The present paper proposes a novel measure for the departure from conditional symmetry based on the cumulative probabilities from the corners of the square table. In a given example, the proposed measure is applied to Japanese occupational status data, and the interpretation of the proposed measure is illustrated as the departure from a proportional structure of social mobility.

The problem of Euler/Tarry revisited

The problem of Euler/Tarry concerning 36 officers can be formulated in mathematical terms: Can a latin square of order 6 have an orthogonal square, or equivalently, are there 6 pairwise disjoint transversals? This was first answered (in the negative) by Tarry (1900/01). We prove the following Theorem: If a latin square of order 6 admits a reflection, i. e. an automorphism of order two which fixes the main diagonal elementwise, then it has no orthogonal square. We list the 12 isomorphism types of latin squares of order 6 and see: they all admit such a reflection. So we get a solution of the Euler problem without the tedious task of tracing the transversals.

Central Diagonal Sections of the n-Cube

We prove that the volume of central hyperplane sections of a unit cube in $\mathbb{R}^n$ orthogonal to a main diagonal of the cube is a strictly monotonically increasing function of the dimension for $n\geq 3$. Our argument uses an integral formula that goes back to Pólya [ 20] (see also [ 14] and [ 3]) for the volume of central sections of the cube and Laplace’s method to estimate the asymptotic behavior of the integral. First, we show that monotonicity holds starting from some specific $n_0$. Then, using interval arithmetic and automatic differentiation, we compute an explicit bound for $n_0$ and check the remaining cases between $3$ and $n_0$ by direct computation.

Algebraic Properties of Parikh Matrices of Binary Picture Arrays

A word is a finite sequence of symbols. Parikh matrix of a word is an upper triangular matrix with ones in the main diagonal and nonnegative integers above the main diagonal which are counts of certain scattered subwords in the word. On the other hand, a picture array, which is a rectangular arrangement of symbols, is an extension of the notion of a word to two dimensions. Parikh matrices associated with a picture array have been introduced, and their properties have been studied. Here, we obtain certain algebraic properties of Parikh matrices of binary picture arrays based on the notions of power, fairness, and a restricted shuffle operator extending the corresponding notions studied in the case of words. We also obtain properties of Parikh matrices of arrays formed by certain geometric operations.

Determinants of Normalized Bohemian Upper Hessenberg Matrices

A matrix is Bohemian if its elements are taken from a finite set of integers. An upper Hessenberg matrix is normalized if all its subdiagonal elements are ones, and hollow if it has only zeros along the main diagonal. All possible determinants of families of normalized and hollow normalized Bohemian upper Hessenberg matrices are enumerated. It is shown that in the case of hollow matrices the maximal determinants are related to a generalization of Fibonacci numbers. Several conjectures recently stated by Corless and Thornton follow from these results.

A Short Note on the Determinant of a Sylvester–Kac Type Matrix

The Sylvester–Kac matrix, also known as Clement matrix, has many extensions and applications. The evaluation of determinant and spectra of many of its generalizations sometimes are hard to compute. Recently, E. Kılıç and T. Arikan proposed an extension the Sylvester–Kac matrix, where the main diagonal is a 2-periodic sequence. They found its determinant using a spectral technique. In this short note, we provide a simple proof for that result by calculating directly the determinant.