Abstract
The matrix equation AX+XA=H where A is symmetric appears in a variety of problems in continuum mechanics and other subjects. Several forms of the solution are available in the literature. We present new solutions which appear to be more concise. This is achieved by employing the adjoint matrix  of A whose elements are the cofactors of A, and by considering the solutions to the symmetric and skew-symmetric parts of H separately. The derivation is no more complicated if we consider the more general matrix equation A T X+XA=H in which A need not be symmetric, and the superscript T denotes the transpose. The main results are shown in (2.6), (3.2), (4.14a,b) and (5.1a,b) for the three-dimensional case and in (6.6a,b) for the two-dimensional case.
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Ting, T.C.T. New expressions for the solution of the matrix equation A T X+XA=H . J Elasticity 45, 61–72 (1996). https://doi.org/10.1007/BF00042471
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DOI: https://doi.org/10.1007/BF00042471