Copenhaver, Martin S.; Kim, Yeon Hyang; Logan, Cortney; Mayfield, Kyanne; Narayan, Sivaram K.; Petro, Matthew J.; Sheperd, Jonathan Diagram vectors and tight frame scaling in finite dimensions. (English) Zbl 1435.42026 Oper. Matrices 8, No. 1, 73-88 (2014). Summary: We consider frames in a finite-dimensional Hilbert space \(\mathcal H_n\) where frames are exactly the spanning sets of the vector space. The diagram vector of a vector in \(\mathbb R^2\) was previously defined using polar coordinates and was used to characterize tight frames in \(\mathbb R^2\) in a geometric fashion. Reformulating the definition of a diagram vector in \(\mathbb R^2\) we provide a natural extension of this notion to \(\mathbb R^n\) and \( \mathbb C^n\). Using the diagram vectors we give a characterization of tight frames in \(\mathbb R^n\) or \(\mathbb C^n\). Further we provide a characterization of when a unit-norm frame in \(\mathbb R^n\) or \(\mathbb C^n\) 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. Cited in 12 Documents MSC: 42C15 General harmonic expansions, frames 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 15A03 Vector spaces, linear dependence, rank, lineability Keywords:frames; tight frames; tight frame scaling; diagram vectors; Gramian operator PDFBibTeX XMLCite \textit{M. S. Copenhaver} et al., Oper. Matrices 8, No. 1, 73--88 (2014; Zbl 1435.42026) Full Text: DOI arXiv Link