Convex invertible sets and matrix sign function

Abstract

It is proved that in two-dimensional unital real algebras a step-by-step construction of minimal convex invertible sets terminates in two steps. An analogous construction of minimal convex conditionally invertible sets of the complex field also terminates in two steps, and the complex field is the only finite dimensional complex algebra with this property. Connections are made with the matrix sign function. The matrix sign function of a matrix belongs to the convex invertible set generated by the matrix. An example is given to show that the matrix sign function cannot always be obtained in two steps, by following the step-by-step construction.