Three Theorems on Vandermond Matrices

Artisevich, A. E. , Shabat, A. B.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 1.

Abstract:
We consider algebraic questions related to the discrete Fourier transform defined using symmetric Vandermonde matrices \(\Lambda\). The main attention in the first two theorems is given to the development of independent formulations of the size \(N\times N\) of the matrix \(\Lambda\) and explicit formulas for the elements of the matrix \(\Lambda\) using the roots of the equation \(\Lambda^N = 1\). The third theorem considers rational functions \(f(\lambda)\), \(\lambda\in \mathbb{C}\), satisfying the condition of "materiality" \(f(\lambda)=f(\frac{1}{\lambda})\), on the entire complex plane and related to the well-known problem of commuting symmetric Vandermonde matrices \(\Lambda\) with (symmetric) three-diagonal matrices \(T\). It is shown that already the first few equations of commutation and the above condition of materiality determine the form of rational functions \(f(\lambda)\) and the equations found for the elements of three-diagonal matrices \(T\) are independent of the order of \(N\) commuting matrices. The obtained equations and the given examples allow us to hypothesize that the considered rational functions are a generalization of Chebyshev polynomials. In a sense, a similar, hypothesis was expressed recently published in "Teoreticheskaya i Matematicheskaya Fizika" by V. M. Bukhstaber et al., where applications of these generalizations are discussed in modern mathematical physics.

Keywords: Vandermond matrix, discrete Fourier transform, commutation conditions, Laurent polynomials.

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