A novel approach to the solution of the tensor equation AX+XA=H

A systematic approach to the solution of the tensor equation AX+XA=H, where A is symmetric, is presented. It is based upon the reformulation of the or iginal equation in the form AX=H where A=Ax1 + 1xA is the fourth-order tens or obtained from the square tensor product of the second-order tensors A an d 1. It is shown that the solution X, which is known to be an isotropic fun ction of A and H, can be effectively obtained either by providing explicit formulas for A(-1) or by reconverting to the format AX=H the well-known rep resentation formulas for tensor-valued isotropic functions. The final form of the solution can thus be established a priori by suitably choosing a set of independent generators for A(-1). The coefficients of the expansion of A(-1) with respect to the assigned generators are then obtained by means of basic composition rules for square tensor products. In this way it is poss ible to provide new expressions of the solution as well as to derive the ex isting ones ia a simpler way. Both three-dimensional and two-dimensional ca ses are addressed in detail. (C) 2000 Elsevier Science Ltd. All fights rese rved.