The relationship between Liouville's arithmetic identities and products of Lambert series is investigated. For example it is shown that Liouville's arithmetic formula for the sumSigma (F(a - b) - F(a + b)), (a,b,x,y)epsilon N-4 ax+by=nwhere n epsilon N and F : Z -> C is an even function, is equivalent to the Lambert series for(Sigma(infinity)(n=1) 1 - q(n)/q(n) sin n theta)(2) (theta epsilon R, |q| < 1)given by Ramanujan.

We solve an inverse symmetric generalized eigenvalue problem Ay = lambda By where B is a given non-singular indefinite finite real symmetric matrix, and A is a positive definite symmetric matrix whose existence is sought, once additional spectral information is appended. We show that there exist an, essentially unique, positive tridiagonal matrix A which solves this problem.