The Expected Norm of Random Matrices

Abstract

We compare the Euclidean operator norm of a random matrix with the Euclidean norm of its rows and columns. In the first part of this paper, we show that if A is a random matrix with i.i.d. zero mean entries, then E∥A∥h ≤ Kh(E maxi ∥ai•∥h + E maxj ∥a•j∥h), where K is a constant which does not depend on the dimensions or distribution of A (h, however, does depend on the dimensions). In the second part we drop the assumption that the entries of A are i.i.d. We therefore consider the Euclidean operator norm of a random matrix, A, obtained from a (non-random) matrix by randomizing the signs of the matrix's entries. We show that in this case, the best inequality possible (up to a multiplicative constant) is E∥A∥h ≤ (c log1/4 min {m, n})h(E maxi ∥ai•∥h + E maxj ∥a•j∥h) (m, n the dimensions of the matrix and c a constant independent of m, n).