Eigenvalue Results for Large Scale Random Vandermonde Matrices With Unit Complex Entries

Abstract

This paper centers on the limit eigenvalue distribution for random Vandermonde matrices with unit magnitude complex entries. The phases of the entries are chosen independently and identically distributed from the interval [−π, π]. Various types of distribution for the phase are considered and we establish the existence of the empirical eigenvalue distribution in the large matrix limit on a wide range of cases. The rate of growth of the maximum eigenvalue is examined and shown to be no greater than O(log N ) and no slower than O(log N/ log log N ) where N is the dimension of the matrix. Additional results include the existence of the capacity of the Vandermonde channel (limit integral for the expected log determinant).

Partially random matrices in line-of-sight wireless networks

“…Besides, numerical simulations suggest that more precise estimates on both the number of significant eigenvalues of GG * /n and its maximal eigenvalue could probably be obtained. Finally, let us mention that the matrix G studied in this paper is related to Vandermonde matrices and random DFT matrices that appear in other contexts in the literature on wireless communications [7], [8], [9], [10] and compressed sensing [11], [12], respectively.…”

“…A minute's thought shows that we can reduce to the case where all the z variables are distinct. Observe that if i pairs of y variables are identified, there will be of the order of n 2l−i terms in the sum of (8), and so to get the same bound as in the previous section, we can afford to bound i of the z variables by 1 in the original integral (9). We now introduce a combinatorial way to look at the problem.…”

Asymptotic results on generalized Vandermonde matrices and their extreme eigenvalues

“…In particular, the limit of the moments of V * V was proved and a combinatorial formula for the asymptotic moments was given under the hypothesis of continuous density. In [1] their results were extended to more general densities and it was also proved that these moments arise as the moments of a probability measure µ ν,β supported on [0, ∞). This measure depends of course on the measure ν, the distribution of the phases, and on the constant β.…”

“…(see [2] or [1] for more details). A random Vandermonde matrix is produced if the entries of the phase vector θ := (θ 1 , .…”

“…The case d = 1 are the matrices in (1). For d ≥ 2, these matrices are defined by selecting L random vectors x q independently in the d-dimensional hypercube It is easy to see that this function is a bijection over the set 0, 1, .…”