Sign Pattern Matrices to Allow Algebraic Positivity

Recent attention in the study of sign pattern matrices is the characterization of sign pattern matrices that require algebraic positivity and the characterization of sign pattern matrices that allow algebraic positivity.In this book we study the conditions for sign pattern matrices of order 2 and 3 to allow algebraic positivity and apply them to the sign controllability of linear control systems.We first give a necessary condition for sign pattern matrices to allow algebraic positivity and based on this fact characterize all spectrally arbitrary sign pattern matrices that allow algebraic positivity.We also prove some conditions for a minimally AP-irreducible sign pattern matrix subpattern of AP-irreducible sign pattern matrix to be minimally irreducible and based on these facts characterize the sign pattern matrices of order 2 and 3 to allow algebraic positivity.The last we define a linear control system to allow algebraic positivity and discuss its relation to a linear control system to require all real eigenvalues.We also prove some conditions for the linear control system to allow algebraic positivity to be sign controllable.