Functional Calculi, Positivity, and Convolution of Matrices
Javad Mashreghi, Mostafa Nasri, Prateek Kumar Vishwakarma
TL;DR
This paper develops a convolution-based analogue of the functional and entrywise calculi for matrices, introducing matrix transforms that respect matrix convolution $A \diamond B$ and preserve positive semidefiniteness. A Cayley–Hamilton-type theorem for the Jury product shows the degree is $M+N-1$ with the annihilating polynomial $p_A(z)=(z-a_{00})^{M+N-1}$, and a novel polynomial-matrix identity for convolution is established. The authors prove positivity preservers analogous to Schoenberg, Rudin, and Horn in the convolution setting, and connect the structure to the Bruhat order on the symmetric group. The results extend the classical theory to the convolutional setting and have potential implications in high-dimensional matrix analysis and discrete probability.
Abstract
Convolution admits a natural formulation as a functional operation on matrices. Motivated by the functional and entrywise calculi, this leads to a framework in which convolution defines a matrix transform that preserves positivity. Within this setting, we establish results parallel to the classical theories of Pólya--Szegő, Schoenberg, Rudin, Loewner, and Horn in the context of entrywise calculus. The structure of our transform is governed by a Cayley--Hamilton-type theory valid in commutative rings of characteristic zero, together with a novel polynomial-matrix identity specific to convolution. Beyond these analytic aspects, we uncover an intrinsic connection between convolution and the Bruhat order on the symmetric group, illuminating the combinatorial aspect of this functional operation. This work extends the classical theory of entrywise positivity preservers and operator monotone functions into the convolutional setting.