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Peer-reviewedCC BY 4.0Mathematics

Completion of Equiangular Tight Frames via QR Decomposition

  • Jun Ji1
  • Long Wang1
  1. 1Department of Mathematics, Kennesaw State University, Marietta, GA 30060, USA
Open at sourceDOICite

Abstract

In this paper, we give a constructive proof of the well-known fact that every ETF of n+1 vectors in Rn equals a signed orthogonal transform of the standard MB frame, establishing an explicit relationship between equiangular tight frames of n+1 vectors in Rn and the standard MB frame via QR factorization. This relationship is utilized to develop a QR-based method for replacing the missing vectors in an equiangular tight frame of n+1 vectors for Rn to complete an equiangular tight frame. A simple recursion and explicit expressions for the entries of the standard MB frame in Rn are also presented. An example in R5 is provided to demonstrate the approach. The preliminary numerical test on randomly generated problems indicates that the method is accurate and stable.

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Cite

BibTeXRIS
BibTeX
@article{ji2026completion,
  title = {{Completion of Equiangular Tight Frames via QR Decomposition}},
  author = {Jun Ji and Long Wang},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14193570},
  url = {https://www.mdpi.com/2227-7390/14/19/3570},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Completion of Equiangular Tight Frames via QR Decomposition
AU  - Jun Ji
AU  - Long Wang
PY  - 2026
DA  - 2026/10/02
JO  - Mathematics
DO  - 10.3390/math14193570
UR  - https://www.mdpi.com/2227-7390/14/19/3570
AB  - In this paper, we give a constructive proof of the well-known fact that every ETF of n+1 vectors in Rn equals a signed orthogonal transform of the standard MB frame, establishing an explicit relationship between equiangular tight frames of n+1 vectors in Rn and the standard MB frame via QR factorization. This relationship is utilized to develop a QR-based method for replacing the missing vectors in an equiangular tight frame of n+1 vectors for Rn to complete an equiangular tight frame. A simple recursion and explicit expressions for the entries of the standard MB frame in Rn are also presented. An example in R5 is provided to demonstrate the approach. The preliminary numerical test on randomly generated problems indicates that the method is accurate and stable.
SN  - 2227-7390
ER  -