A key step in the computation of the unitary dual of a Lie group is the determination if certain rational symmetric matrices are positive semidefinite. The size of some of the computations dictates that high performance integer matrix computations be used. We explore the feasibility of this approach by developing three algorithms for integer symmetric matrix signature and studying their performance both asymptotically and by experimenting on a particular matrix family constructed from the exceptional Weyl group E8. We conclude that the computation is doable, with a parallel implementation needed for the largest representations.