The quasi principal rank characteristic sequence

Abstract 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 q 1 q 2 ⋯ q n , where q k 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.