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Estimating the condition numbers of random structured matrices is a wellknown challenge, linked to the design of efficient randomized matrixalgorithms. We deduce such estimates for Gaussian random Toeplitz and circulantmatrices. The former estimates can be surprising because the condition numbersgrow exponentially in n as n grows to infinity for some large and importantclasses of n-by-n Toeplitz matrices, whereas we prove the opposit for Gaussianrandom Toeplitz matrices. Our formal estimates are in good accordance with ournumerical tests, except that circulant matrices tend to be even betterconditioned according to the tests than according to our formal study.

Condition Numbers of Random Toeplitz and Circulant Matrices