11. (Gasca-Pena) 一个 $n$ 阶可逆矩阵 $A$ 是全面非负的当且仅当对每个 $1\leq k\leq n$, $$\bex \det A[1,2,\cdots,k]>0, \eex$$ $$\bex \det A[\al\mid 1,2,\cdots,k]\geq 0,\quad \det A[1,2,\cdots,k\mid \al]\geq 0,\quad \forall\ \al\in Q_{k,n}. \eex$$
证明: 见 [M. Gasca, J.M. Pe\~na, Total positivity, $QR$ factorization, and Neville elimination, SIAM J. Matrix Anal. Appl., 14 (1993), 1132--1140] 定理 3.1.
