Least Squares Sign-Solvability
Bryan L. Shader · SIAM Journal on Matrix Analysis and Applications · 1995
Let $Ax = b$ be a linear system. We study the relationship between the sign pattern of the least squares solution to $Ax = b$ and the sign patterns of A and b. The system $Ax = b$ is least squares sign-solvable if the signs of the entries in its least square solution can be determined solely from the signs of the entries of A and b. We construct a family of least squares sign-solvable linear systems from the vertex-incidence matrices of trees. General properties of least squares sign-solvable linear systems are developed and the structure of a least squares sign-solvable system is shown to be analogous to that of sign-solvable linear systems. Square matrices whose sign pattern determines the sign pattern of its inverse have been extensively studied. We study matrices whose sign pattern determines the sign pattern of its generalized inverse.