The principal rank characteristic sequence over various fields

Given an n×nn×n matrix, its principal rank characteristic sequence is a sequence of length n+1n+1 of 0s and 1s where, for k=0,1,…,nk=0,1,…,n, a 1 in the kth position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix. The principal rank characteristic sequences for symmetric matrices over various fields are investigated, with all such attainable sequences determined for all n over any field with characteristic 2. A complete list of attainable sequences for real symmetric matrices of order 7 is reported.