Among all the ways one might define f(A) for a square complex matrix A and a given function f:C→C, the notion of a primary matrix function is perhaps the most useful and natural. Using only basic conceptual properties of primary matrix functions, we consider whether f(A) can have rank one, and whether, for a given B, there is a unique A such that f(A)=B. © 2002 Elsevier Science Inc. All rights reserved.
Horn, R. A., & Piepmeyer, G. G. (2003). Two applications of the theory of primary matrix functions. In Linear Algebra and Its Applications (Vol. 361, pp. 99–106). https://doi.org/10.1016/S0024-3795(02)00267-7