We combine some basic techniques of linear algebra with some expressions for Toeplitz and circulant matrices and the properties of Gaussian random matrices to estimate the norms of Gaussian Toeplitz and circulant random matrices and their inverses. In the case of circulant matrices we obtain sharp probabilistic estimates, which show that the matrices are expected to be very well conditioned. Our probabilistic estimates for the norms of standard Gaussian Toeplitz random matrices are within a factor of √2 from those in the circulant case. We also achieve partial progress in estimating the norm of the Toeplitz inverse. Namely we yield reasonable probabilistic upper estimates assuming certain bounds on the absolute values of two corner entries of the inverse. Empirically we observe that the condition numbers of Toeplitz and general random matrices tend to be of the same order. As the matrix size grows, these numbers grow equally slowly, although faster than in the case of circulant random matrices.
Pan, V. Y., Svadlenka, J., & Zhao, L. (2015). Estimating the norms of random circulant and Toeplitz matrices and their inverses. Linear Algebra and Its Applications, 468, 197–210. https://doi.org/10.1016/j.laa.2014.06.027