Given a real n ×n matrix A_n, we compute the expectation and the variance of the random variable || A_n x||² where x is uniformly distributed on the unit sphere of the Euclidean Rⁿ. Combining the result with results by Avram, Parter, and Tyrtyshnikov on singular value distribution, we can show that if A_n = T_n(b) is a Toeplitz matrix and n is large, then || T_n(b)x || / (||T_n(b)|| ||x||) clusters fairly sharp around the constant || b||₂/||b||_¥. We discuss applications of this observation to structured normwise condition numbers, and we also embark on the behavior of the distance of a large Toeplitz band matrix to the nearest singular matrix.

Footnotes:

¹ Joint work with S. GRUDSKY.