Diagram vectors and Tight Frame Scaling in Finite Dimensions

Abstract

We consider frames in a finite-dimensional Hilbert space Hn where frames are exactly the spanning sets of the vector space. The diagram vector of a vector in R2 was previously defined using polar coordinates and was used to characterize tight frames in R2 in a geometric fashion. Reformulating the definition of a diagram vector in R2 we provide a natural extension of this notion to Rn and Cn. Using the diagram vectors we give a characterization of tight frames in Rn or Cn. Further we provide a characterization of when a unit-norm frame in Rn or Cn can be scaled to a tight frame. This classification allows us to determine all scaling coefficients that make a unit-norm frame into a tight frame.