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.
New factorization results dealing mainly with P -matrices and M -matrices are presented. It is proved that any matrix in Mn(R) with positive determinant can be written as the product of three P -matrices (compared with the classical result that five positive definite matrices may be needed). It is also proved that a matrix A with positive determinant can be stabilized via multiplication by a P -matrix if and only if A is not a diagonal matrix with all diagonal entries negative. Factorization into two P -matrices is considered and characterized for n = 2. Using elementary bidiagonal factorization results, it is shown that the nonsingular M -matrices, or the nonsingular totally nonnegative matrices, generate all matrices in Mn(R) with positive determinant. Further results on products of M -matrices and inverse M -matrices are given.
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