\C-flows A^z of linear maps A expressed in terms of A^{-1},A^{-2},...,A^{-n} and analytic functions of z

TL;DR

This paper develops a method to express complex powers of a matrix A as a linear combination of its integer powers with analytic coefficient functions, extending the classical polynomial relation to a continuous parameter z.

Contribution

It introduces a construction of analytic functions c_i(z) for matrix powers A^z, generalizing polynomial relations to complex exponents using companion matrices.

Findings

Provides explicit formulas for c_i(z) in terms of matrix C and vector α.

Shows how to extend polynomial relations to complex powers of matrices.

Offers a new perspective on matrix functions and their analytic representations.

Abstract

Suppose A\in GL_n(\C) has a relation A^p=c_{p-1}A^{p-1}+.... + c_1 A+ c_0I where the c_i in \C. This article describes how to construct analytic functions c_i(z) such that A^z=c_{p-1}(z)A^{p-1}+... + c_1(z) A+ c_0(z)I . One of the theorems gives a possible description of the c_i(z): c_i(z)=C^z\alpha where C\in Mat_p(\C) is (similar to) the companion matrix of X^p-c_{p-1}X^{p-1}-... -c_1X-c_0I, and \alpha:= (c_{p-1},...,c_1,c_0)^t.