Abstract

We outline some ideas on multicentric calculus.

The basic technical tool is to change variable z to w as follows:

w = p(z).

Here p is a polynomial in the complex plane (or real line) with simple roots. Since this is a many-to-one mapping, we compensate the “loss of information” by introducing a vector valued function f to represent the original scalar function φ. We call the approach as “multicentric calculus”.

Multicentric calculus can be used in an effective way to compute φ(A) when p(A) is much “nicer” than A, in particular, when an effective functional calculus exists for p(A) but is not directly available for A. We discuss two directions, one based on holomorphic functional calculus. The other direction then generalizes the simple calculus where continuous functions are evaluated at diagonalizable matrices as follows:

φ(A) = Tφ(D)T−1.

In particular, one need not assume that φ is differentiable if the matrix A has nontrivial Jordan blocks. This is so far unpublished.