The Expected Norm of Random Matrices

Source 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 [les ] K h ( E max i ∥ a i [bull ] ∥ h + E max j ∥ a j [bull ] ∥ 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 [les ] ( c log 1/4 min { m , n }) h ( E max i ∥ a i [bull ] ∥ h + E max j ∥ a j [bull ] ∥ h ) ( m , n the dimensions of the matrix and c a constant independent of m , n ).