Equivalence Classes of Parseval Frames

On the set of frames in a finite-dimensional space, we introduce the widest possible equivalence preserving the main characteristics of frames, that is, tightness, equiangularity, and spark (the least number of linearly dependent vectors). This equivalence is known as projective–permutational unitary equivalence. For example, we show that full spark equiangular tight frames in the spaces $\mathbb{R}^3$, $\mathbb{R}^5$, and $\mathbb{R}^7$ are unique up to equivalence. A similar uniqueness result is obtained for the general uniform Parseval frame of $d+1$ vectors in the space $\mathbb{R}^d$. Related questions have been raised in the literature several times. Calculating the spark is computationally much harder than calculating the rank of a matrix. Here we present an algorithm that can possibly simplify the spark calculation. The use of Seidel matrices and the Naimark complement technique proves to be very useful in the classification of frames up to equivalence.