The spectrum of symmetric Krawtchouk matrices
Define for a natural number \(N\) a symmetric matrix \(N + 1 \times N + 1\) matrix \(S\) via the equation \(\sum_{ij} S_{ij} x^i y^j = (1 + x + y - xy)^N\). Then \(S_{ij}\) equals \({n \choose j}\) times \(\Phi_{ij}\), the latter being the value of the \(i\)th Krawtchouk polynomial evaluated at \(j\). As a contribution to the understanding of Krawtchouk polynomials -- these occur in such varied areas as quantum mechanics, coding theory, random variables, representation theory, orthogonal polynomials -- the authors provide partial information on spectrum and eigenvectors of the symmetrized version \(S\) of the Krawtchouk matrix \(\Phi \). First they establish a number of commutation properties for various (tri)diagonal matrices, reflections, and \(\Phi\), and cite a Lie group-theoretical result according to which matrices \(\Phi\) are essentially similar via a certain exponential to a particular reflection. Their generating function formulation for \(S\) allows them then to apply Fourier analysis on the unit circle; being a further tool an extended Perron Frobenius theorem (as given by \textit{H. Minc} [Nonnegative Matrices, Wiley (1988; Zbl 0638.15008)], they prove the following: Theorem. If \(N\) is even, \(N = 2M\) say, then the vector whose \(2k\)-th coordinate is \({2M - 2k \choose M - k} {2k \choose k}\) for \(k = 0, \dots, M\) and 0 otherwise, is an eigenvector of \(S\) with eigenvalue \(\text{tr} S = {N \choose M}\); furthermore \(\text{tr} S\) is a simple root of the characteristic polynomial of \(S\) and eigenvalue of minimal modulus. Tables and indicationss for generating matrices \(\Phi\) and \(S\) are appended.
scientific article; zbMATH DE number 871115