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 satis…es 2 (A) max i2[m] P j2[n] jaij j cj max aij 6=0 ricj ; where ri = P k2[n] jaikj ; cj...

Abstract. We study the q-singular values of random matrices with preGaussian entries defined in terms of the q-quasinorm with 0 < q ≤ 1. In this paper, we mainly consider the decay of the lower and upper tail probabilities of the largest q-singular value s 1 , when the number of rows of the matrices becomes very large. Based on the results in probabilistic estimates on the largest q-singular va...