The quasi principal rank characteristic sequence

A minor of a matrix is quasi-principal if it is a principal or an almost-principal minor. The quasi principal rank characteristic sequence (qpr-sequence) of an n×n symmetric matrix is introduced, which is defined as q1q2⋯qn, where qk is A, S, or N, according as all, some but not all, or none of its quasi-principal minors of order k are nonzero. This sequence extends the principal rank characteristic sequences in the literature, which only depend on the principal minors of the matrix. A necessary condition for the attainability of a qpr-sequence is established. Using probabilistic techniques, a complete characterization of the qpr-sequences that are attainable by symmetric matrices over fields of characteristic 0 is given.