Around a multivariate Schmidt-Spitzer theorem

作者：Alexandersson Per^*; Shapiro Boris

来源：Linear Algebra and Its Applications, 2014, 446: 356-368.

DOI：10.1016/j.laa.2014.01.005

• Asymptotic root distribution
• Square and rectangular Toeplitz matrices

Given an arbitrary complex-valued infinite matrix A = (a(upsilon j)), i = 1,..., infinity; j = 1,., infinity and a positive integer n we introduce a naturally associated polynomial basis B-A of C[x(0),..., x(n)]. We discuss some properties of the locus of common zeros of all polynomials in B-A having a given degree m; the latter locus can be interpreted as the spectrum of the m x (m + n)-submatrix of A formed by its m first rows and by the (m + n) first columns We initiate the study of the asymptotic of these spectra when m -> infinity n in the case when A is a banded Toeplitz matrix. In particular, we present and partially prove a conjectural multivariate analog of the well-known Schmidt-Spitzer theorem which describes the spectral asymptotic for the sequence of principal minors of an arbitrary banded Toeplitz matrix. Finally, we discuss relations between polynomial bases B-A and multivariate orthogonal polynomials.

• 出版日期2014-4-1