On the almost-principal minors of a symmetric matrix

The almost-principal rank characteristic sequence (apr-sequence) of an $n\times n$ symmetric matrix is introduced, which is defined to be the string $a_1 a_2 \cdots a_{n-1}$, where $a_k$ is either $\tt A$, $\tt S$, or $\tt N$, according as all, some but not all, or none of its almost-principal minors of order $k$ are nonzero. In contrast to the other principal rank characteristic sequences in the literature, the apr-sequence of a matrix does not depend on principal minors. The almost-principal rank of a symmetric matrix $B$, denoted by ${\rm aprank}(B)$, is defined as the size of a largest nonsingular almost-principal submatrix of $B$. A complete characterization of the sequences not containing an $\tt A$ that can be realized as the apr-sequence of a symmetric matrix over a field $\mathbb{F}$ is provided. A necessary condition for a sequence to be the apr-sequence of a symmetric matrix over a field $\mathbb{F}$ is presented. It is shown that if $B \in \mathbb{F}^{n\times n}$ is symmetric and non-diagonal, then ${\rm rank}(B)-1 \leq {\rm aprank}(B) \leq {\rm rank}(B)$, with both bounds being sharp. Moreover, it is shown that if $B$ is symmetric, non-diagonal and singular, and does not contain a zero row, then ${\rm rank}(B) = {\rm aprank}(B)$.