Revisiting Schur ’ s bound on the largest singular value Vladimir Nikiforov

We give upper and lower bounds on the largest singular value of a matrix using analogues to walks in graphs. For nonnegative matrices these bounds are asymptotically tight. In particular, the following result improves a bound due to Schur. If A = (aij) is an m× n complex matrix, its largest singular value satisfies σ (A) ≤ max i∈[m] ∑ j∈[n] |aij| cj ≤ max aij 6=0 ricj, where ri = ∑ k∈[n] |aik| , cj = ∑ k∈[m] |akj| .