Minor-Order Exponent Profiles of Oscillatory Matrices

Source abstract

For an $n\times n$ oscillatory matrix $A$, let $e_k(A)$ be the least positive integer $m$ for which every minor of order $k$ of $A^m$ is positive. We determine the profile $(e_1(A),\ldots,e_n(A))$. For every nonsingular totally nonnegative matrix, the column index sets of positive entries in each compound row form an interval in the componentwise order. The endpoint maps are order-preserving and compose under multiplication. This yields a two-corner criterion for positivity of all minors of a fixed order. Consequently, $e_k(A)$ is the larger of the first positivity times of two remote corner minors. We prove the sharp inequalities $e_k(A)\leq\max{k,n-k}$ for $1\leq k<n$ and $|e_{k+1}(A)-e_k(A)|\leq1$ for $1\leq k\leq n-2$. For $D=\operatorname{diag}(1,-1,1,-1,\ldots)$, the matrix $DA^{-1}D$ is oscillatory and satisfies $e_k(DA^{-1}D)=e_{n-k}(A)$ for $1\leq k<n$; the determinant exponent remains one. The ordered positions of the positive factors in an adjacent bidiagonal factorization determine the profile independently of their values. Each one-sided factor sequence can be represented by a single permutation, giving a finite characterization of all profiles and an integer unimodular realization of each. We also obtain a compatibility condition across minor orders: if $3\leq k\leq n-3$ and $e_k(A)\leq2$, then $e_j(A)\leq2+\lceil |j-k|/2\rceil$ for $2\leq j\leq n-2$. In particular, $e_k(A)\leq2$ implies $e_{k+2}(A)\leq3$ for $3\leq k\leq n-4$, and both the constant and this range are sharp.