Alexander Belton (University of Plymouth)
Date: Wednesday 17 May 2023, 15:00 (face-to-face)
Venue: Rolle 116, University of Plymouth
Title: Entrywise Preservers for Classes of Positive Matrices
Abstract: We all know the correct way to multiply matrices, but it is also possible to treat them as simple arrays of numbers and perform algebraic operations entry by entry. For multiplication, this is called the Hadamard product.
It may seem surprising but the Hadamard product preserves the collection of matrices that are positive semidefinite: those real symmetric matrices with non-negative eigenvalues. It follows immediately that applying any absolutely monotonic function entrywise also preserves this form of positivity. (A function is absolutely monotonic if its Maclaurin series has non-negative coefficients). Rather more work is required to show that the converse is true: a function which preserves positive semidefiniteness when applied entrywise to matrices of arbitrary size is necessarily absolutely monotonic.
The situation is more complex for matrices of a fixed size, or when the class of matrices under study has some other form of positivity or possesses additional structure, such as Hankel or Toeplitz matrices. This talk will discuss results for some of these situations.
This is joint work with Dominique Guillot (University of Delaware), Apoorva Khare (Indian Institute of Science, Bangalore) and Mihai Putinar (University of California at Santa Barbara and Newcastle University).