Diagram vectors and tight frame scaling in finite dimensions

We consider frames in a finite-dimensional Hilbert space H_n 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.