THE TENSOR EQUATION AX+XA=PHI(A,H), WITH APPLICATIONS TO KINEMATICS OF CONTINUA

The (second-order) tensor equation AX + XA = PHI(A, H) is studied for certain isotropic functions PHI(A, H) which are linear in H. Qualitati ve properties of the solution X and relations between the solutions fo r various forms of PHI are established for an inner product space of a rbitrary dimension. These results, together with Rivlin's identities f or tensor polynomials in two variables, are applied in three dimension s to obtain new explicit formulas for X in direct tensor notation as w ell as new derivations of previously known formulas. Several applicati ons to the kinematics of continua are considered.