Speaker
Description
Classical approaches to matrix function theory — i.e., extending scalar functions to matrices — are largely organized around two frameworks: entrywise calculus via the Schur (Hadamard) product, and functional calculus via the spectral theorem. In this talk, I present a third, fundamentally different framework based on matrix convolution, in which convolution itself is viewed as a matrix multiplication and gives rise to a new class of matrix transforms.
This perspective leads to a systematic theory of convolution-based transforms and their positivity preserving properties. I will describe convolutional analogues of the classical results of Pólya–Szegő, Schoenberg, Rudin, Loewner, and Horn, including a Schoenberg-type classification theorem showing that absolute monotonicity is again the correct notion for positivity preservers across all dimensions, together with refined results in fixed matrix dimensions.
Beyond positivity, I will highlight two structural features that distinguish convolution from both standard and entrywise products: a Cayley–Hamilton-type theorem for convolution with optimal degree, and a novel polynomial–matrix identity underlying the associated transforms. If time permits, I will also briefly describe an unexpected connection with the Bruhat order on the symmetric group, and an application to sums of discrete random variables, illustrating new links between matrix analysis, algebraic combinatorics, and probability.
(Based on joint work with Javad Mashreghi and Mostafa Nasri.)