Derivatives of triangular, Toeplitz, circulant matrices and matrices of other forms over semirings

TL;DR

This paper explores the construction and properties of derivations in matrix semirings, focusing on various matrix forms such as diagonal, triangular, Toeplitz, and circulant matrices, and establishing related theorems.

Contribution

It introduces new examples of derivations in matrix semirings and provides theoretical results on derivatives of specific matrix classes.

Findings

Constructed examples of derivations in matrix semirings.

Proved theorems for derivatives of various matrix forms.

Analyzed hereditary and inner derivations.

Abstract

In this article we construct examples of derivations in matrix semirings. We study hereditary and inner derivations, derivatives of diagonal, triangular, Toeplitz, circulant matrices and of matrices of other forms and prove theorems for derivatives of matrices of these forms.