Total nonnegativity of a matrix

A real rectangular matrix is totally nonnegative if every square minor, with row and column indices in increasing order, is nonnegative. A nonnegative bidiagonal matrix is totally nonnegative, and products of compatible totally nonnegative matrices have the same property by the Cauchy–Binet formula. This controls signs of inverse entries and underlies stability and shape-preserving properties of B-spline collocation. Total nonnegativity permits zero minors; strict total positivity requires every such minor to be positive.

If an invertible square totally nonnegative matrix $A$ has size $n$, then $\det A>0$ andIndeed the adjugate identity gives $(A^{-1}{)}_{ij}=(-1{)}^{i+j}\det A_{\hat{j},\hat{i}}/\det A$, and the deleted-row/deleted-column minor is nonnegative. Consequently an alternating vector $v_j=(-1{)}^j$ gives $(A^{-1}v{)}_i=(-1{)}^i{\sum }_j|(A^{-1}{)}_{ij}|$.