This question is closely related to some questions I already asked
Given a tuple of positive integers (such as (1,2,5) ), is there a matrix A with positive integer entries such that the integers in the tuple are its eigenvalues (In our example, the matrix [ [2,1,1] , [1,2,1] , [1,1,4] ] does the job) ? I am only interested in tuples with distinct values.
Note, that the answer is trivial if zeros are allowed.
I came across the useful perron-frobenius-theorem, which rules out many possible tuples. The main question is : Is there any number n such that there is a nxn-matrix A with positive integers with the eigenvalues 1,...,n ?
For 2x2-matrices I have checked that every tuple [m,n] with n > m + 1 is possible. For 3x3-matrices I have checked that the lowest possible value for the largest eigenvalue is 5. For 4x4-matrices, my record for the lowest largest eigenvalue is 7, for 5x5 it is 8. A nice answer to an earlier question proves that for every n there is a matrix with largest eigenvalue 2n, but there is still room for the optimal.