Full-rank characterization of generalized Vandermonde matrices

Prove that a generalized Vandermonde matrix with distinct points on the unit circle and strictly positive, distinct, potentially non-integer powers has rank equal to the minimum of its number of rows and columns with probability one.

Background

The paper represents the temporal component of non-uniform delay-coordinate maps for linear dynamical systems using generalized Vandermonde matrices whose powers are distinct, positive, and unevenly spaced. Unlike ordinary Vandermonde matrices, these matrices do not generally admit a determinant formula based on polynomial interpolation because their powers may be non-integer and non-consecutive.

The authors establish the conjectured rank property only in limited cases, including small matrix dimensions, and derive asymptotic and numerical evidence for selected random sampling schemes. The conjecture is required to justify the claimed stable Takens embedding theorem for non-uniformly sampled linear systems.

References

Thus, we state the following, only as a conjecture, about the rank of a generalized Vandermonde matrix. Let $T_{m,\ell} \in C{\ell \times m}$ be as eq:Generalized_Vandermonde_Matrix. Let ${\tau_k}{k = 1}\ell$ be a set of strictly-positive, distinct powers in the sense that $\tau_a \neq \tau_b$ for any $a,b \in {1,...,\ell}$, and let ${z_p}{p = 1}m \subseteq S1$ be a distinct set points with associated angles ${\theta_p}{p = 1}m$ in the interval $(-\pi,\pi]$ in the sense that $\theta_p \neq \theta_q$ for $p,q \in {1,...,m}$. Then with probability $1$, $\rank(T{m,\ell}) = \min{m,\ell}$.

— Stable Takens' Embedding Theorem for Non-Uniformly-Sampled Linear Systems  (2608.14001 - Ng et al., 14 Aug 2026) in Conjecture 1, Section III, “Generalized Vandermonde Matrices and Times Series”