 
Summary: COLLOQUIUM
University of Regina
Department of Mathematics and Statistics
Speaker: Shaun Fallat (University of Regina)
Title: Sign Variation of Vectors and Eigenvalues of Oscillatory Matrices
Time & Place: Friday, September 18, 3:15  4:15 pm, CL 435
Abstract
In 1930, I. Schoenberg proved the seminal result that, as a linear transfor
mation, a totally positive matrix cannot increase the number of sign changes
in a vector. In fact, the transformations that cannot increase the number
of sign variations are of interest in a variety of applications, including ap
proximation theory and shape preserving functions.
Around the same time, but from a different perspective, Gantmacher and
Krein proved that the eigenvalues of an oscillatory matrix are positive and
all distinct. In addition, though often less advertised, they proved a num
ber of interesting properties about the associated eigenvectors of oscillatory
matrices.
In this talk, I will review the above facts, and establish a connection
between them culminating in a new, more comprehensive, characterization
of the spectral structure of oscillatory matrices.
