Sign Controllability of a Nonnegative Matrix and a Positive Vector

An n-by-n matrix A and a vector b are controllable if and only if the matrix $[ b,Ab,A^2 b, \ldots ,A^{n - 1} b ]$ has rank n. An array with each entry equal to +, –, or 0 is a sign pattern. If ${\bf A}$ and ${\bf B}$ are sign patterns of orders n-by-n and n-by-1, respectively, then the pair $( {\bf A},{\bf B} )$ is called sign controllable if $( A,b )$ is controllable for all $A \in {\bf A}$ and for all $b \in {\bf B}$ . Sign controllability of a nonnegative sign pattern ${\bf A}$ and a positive sign pattern ${\bf B}$ are characterized, and sufficient conditions for other cases of the sign controllability problem are given.