The Tensor Equation AX + XA = G

We study the second-order tensor equation AX XA G for symmetric, positive-definite A and arbitrary G. Motivated by applications in the continuum mechanics literature, we also examine some special cases where G depends on A and another tensor H. For arbitrary dimensions, we establish relations between the solutions X for various forms of G. These results, together with Rivlins identities for tensor polynomials in two variables, are applied in two and three dimensions to obtain explicit formulas for X in direct component-free notation. The results include new formulas as well as new derivations of previously known formulas. An application to the kinematics of rigid motions is considered.