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- W4289752010 abstract "The almost-principal rank characteristic sequence (apr-sequence) of an $ntimes n$ 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 $B in mathbb{F}^{ntimes n}$ 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 ${rm rank}(B) = {rm aprank}(B)$." @default.
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- W4289752010 date "2018-07-19" @default.
- W4289752010 modified "2023-09-29" @default.
- W4289752010 title "On the almost-principal minors of a symmetric matrix" @default.
- W4289752010 doi "https://doi.org/10.48550/arxiv.1807.07448" @default.
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