The enhanced principal rank characteristic sequence for skew-symmetric matrices

Abstract

The enhanced principal rank characteristic sequence (epr-sequence) was originally defined for an n×n real symmetric matrix or an n×n Hermitian matrix. Such a sequence is defined to be ℓ1ℓ2⋯ℓn where ℓk is A, S, or N depending on whether all, some, or none of the matrix principal minors of order k are nonzero. Here we give a complete characterization of the attainable epr-sequences for real skew-symmetric matrices. With the constraint that ℓk=0 if k is odd, we show that nearly all epr-sequences are attainable by skew-symmetric matrices, which is in contrast to the case of real symmetric or Hermitian matrices for which many epr-sequences are forbidden.