Any submultiplicative norm (for example the row-sum-norm) is an upper bound for the spectralradius of a matrix A. But is there a way to get a suitable LOWER bound for the spectralradius ?
Motivation : I would like to have a lower bound for the eigenvalues of a matrix with positive integer entries. In particular, is there a nxn-matrix A with positive integer entries and the integer eigenvalues 1..n ? The answer for n>1 seems to be no. Even more interesting would be the following : If a nxn-matrix A has positive integer entries and distinct positive integer eigenvalues (such as [ [4,1,1] , [1,2,1] , [1,1,2] ]) what is the least possible value for the largest eigenvalue (in the example it is 5 because the eigenvalues are 1,2 and 5).