A few weeks ago, Ramon van Handel and I uploaded a paper entitled “Sharp non asymptotic bounds on the norm of random matrices with independent entries” to the arxiv. In this blog post, I’ll attempt to describe and motivate the results there.

The main motivation comes from the vast number of applications for bounds on the spectral norm of random matrices, some of which I have described on this blog (see here, here, and here). In many (applied) math problems one needs to control the spectral norm of a certain application-specific random matrix. A common approach, which has found great success, is to introduce randomness in the models and control this quantity on certain random matrices. Perhaps the simplest example of how randomness can help is the spectral norm of an standard Wigner matrix, a symmetric matrix with iid standard gaussian entries. As each entry of the matrix is essentially of constant order the spectral norm of such a matrix can go up to , however, due to a concentration phenomenon, it is almost always roughly . Unfortunately, random matrix in many applications do not correspond to this simple model.

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