Question

This question is related to my earlier question here .

Given an $n\times n$ random matrix $A$, is determining the properties (mean, variance,moments,etc.) of its induced $p$-norm ($p\neq 0,1,2,\infty$) a good research problem?

Looking at related work, I couldn't find anything much except a paper by Hansen on the 2-norm here . Also Terence Tao's page here on random matrices talks about singular values of a random matrix ; I don't know if there is a trivial extension from these results to determining its $p$ -norm.

I don't have a pure maths background and would much appreciate suggestions.

Answer 1

Here are some papers you might want to look at if you're interested in this issue (probably better than the paper of mine that Suvrit mentioned):

1. G. Bennett, V. Goodman, and C.M. Newman. Norms of random matrices. Pacific J. Math. 59 (1975), no. 2, 359–365.

2. Grahame Bennett. Lectures on matrix transformations of $l_p$ spaces. Notes in Banach spaces, pp. 39--80, Univ. Texas Press, Austin, Tex., 1980. (specifically, section 4 of this paper)

3. Stanisław Szarek. Condition numbers of random matrices. J. Complexity 7 (1991), no. 2, 131–149.

4. Kenneth R. Davidson and Stanislaw J. Szarek. Local operator theory, random matrices and Banach spaces. Handbook of the geometry of Banach spaces, Vol. I, 317–366, North-Holland, Amsterdam, 2001. (specifically, the end of section 2.3 of this paper)

The third of these deals explicitly with the question of deducing results about the induced $p$-norm from results about singular values.

Answer 2

I just found a paper by Mark Meckes (I hope he sees your post and gives a more informed answer than mine), that seems to go a long way towards answering your question.

In the following paper: Concentration of norms and eigenvalues of random matrices. J. Funct. Anal. 211 (2004) no. 2, 508-524. (Links: Paper or arXiv), the author proves concentration (around a median) results for $\ell_p$ operator norms of random matrices. The techniques of that paper should help you further.