An approach to isotropic tensor functions and their derivatives via omega matrix calculus.

Abstract

In this work we show how to obtain a closed form expression of any isotropic tensor function F (A) and their associated derivatives with A a second order tensor in a finite dimensional space. Our approach is based on a recent work of the author (SIAM Rev. 62(1):264–280, 2020) extending the Omega operator calculus, originally devised by MacMahon to describe partitions of natural numbers, to the realm of matrix analysis, namely, the Omega Matrix Calculus (OMC). The OMC is conceptually simple and useful in practice. Indeed, we show that the Cayley-Hamilton theorem and an improvement for low-rank second order tensors due to Segercrantz (Am. Math. Mon. 99(1):42–44, 1992), the representation of isotropic tensor functions and their first derivative of Itskov (Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 459(2034):1449–1457, 2003), and Theorem 1.1 of Norris (Q. Appl. Math. 66(4):725–741, 2008) are all special cases of a general Omega expression introduced in this work.