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An n × n sign pattern A is spectrally arbitrary if given any self-conjugate spectrum there exists a matrix realization of A with that spectrum. If replacing any nonzero entry (or entries) of A by zero destroys this property, then A is a minimal spectrally arbitrary sign pattern. For n ≥ 3, several families of n × n spectrally arbitrary sign patterns are presented, and their minimal spectrally arbitrary subpatterns are identified. These are the first known families of n × n minimal spectrally arbitrary sign patterns. Furthermore, all such 3 × 3 sign patterns are determined and it is proved that any irreducible n × n spectrally arbitrary sign pattern must have at least 2n − 1 nonzero entries, and conjectured that the minimum number of nonzero entries is 2n.

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Several necessary or sufficient conditions for a sign pattern to allow eventual positivity are established. It is also shown that certain families of sign patterns do not allow eventual positivity. These results are applied to show that for n ≥ 2, the minimum number of positive entries in an n×n sign pattern that allows eventual positivity is n+1, and to classify all 2×2 and 3×3 sign patterns as to whether or not the pattern allows eventual positivity. A 3 × 3 matrix is presented to demonstrate that the positive part of an eventually positive matrix need not be primitive, answering negatively a question of Johnson and Tarazaga.