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#### Article Title

An improved estimate for the condition number anomaly of univariate Gaussian correlation matrices

#### Keywords

Euclidean distance matrix, Gaussian correlation matrix, almost negative definite matrix, spatial linear model, radial basis functions, Vandermonde matrix, condition number

#### Abstract

In this short note, it is proved that the derivatives of the parametrized univariate Gaussian correlation matrix R_g (θ) = (exp(−θ(x_i − x_j )^2_{i,j} ∈ R^{n×n} are rank-deficient in the limit θ = 0 up to any order m < (n − 1)/2. This result generalizes the rank deficiency theorem for Euclidean distance matrices, which appear as the first-order derivatives of the Gaussian correlation matrices in the limit θ = 0. As a consequence, it is shown that the condition number of R_g(θ) grows at least as fast as 1(/θ^(mˆ +1) for θ → 0, where mˆ is the largest integer such that mˆ < (n − 1)/2. This considerably improves the previously known growth rate estimate of 1/θ^22 for the so-called Gaussian condition number anomaly.

#### Recommended Citation

Zimmermann, Ralf.
(2015),
"An improved estimate for the condition number anomaly of univariate Gaussian correlation matrices",
*Electronic Journal of Linear Algebra*,
Volume 30, pp. 592-598.

DOI: https://doi.org/10.13001/1081-3810.2839

*Abstract*