Nonnegative Solutions of a Quadratic Matrix Equation Arising from Comparison Theorems in Ordinary Differential Equations

We study the quadratic matrix equation \[ X^2 + \beta X + \gamma A = 0, \] where A is a given elementwise nonnegative (resp. positive semi-definite) matrix and the solution X is required to be an elementwise nonnegative (resp. positive semi-definite) matrix. When $\beta = - 1$ and $\gamma = 1$, our results may be used, for example, to obtain a simple nonoscillation criterion for the matrix differential equation \[ Y'' ( t ) + Q ( t )Y ( t ) = 0, \] where Y and Q are matrix-valued functions and ${}^\prime$ denotes differentiation. This generalizes a result of Hille for the scalar case. Extensions are given when A and X are nonnegative with respect to more general cone orderings.