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 original equation in the form AX=H where AˆAut 1 ‡ 1ut A is the fourth- order tensor obtained from the square tensor product of the second-order tensors A and 1. It is shown that the solution X, which is known to be an isotropic function of A and H, can be e€ectively obtained either by providing explicit formulas for A ÿ1 or by reconverting to the format AX=H the well-known representation formulas for tensor-valued isotropic functions. The ®nal form of the solution can thus be established a priori by suitably choosing a set of independent generators for A ÿ1. The coecients of the expansion of A ÿ1 with respect to the assigned generators are then obtained by means of basic composition rules for square tensor products. In this way it is possible to provide new expressions of the solution as well as to derive the existing ones in a simpler way. Both three-dimensional and two-dimensional cases are addressed in detail.