New expressions for the solution of the matrix equation A T X+XA=H

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 se.,parately. The derivation is no more complicated if we consider the more gener.al matrix equation A" 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. la,b) for the three-dimensional case and in (6.6a,b) for the two-dimensional case.

Some basis-free expressions for stresses conjugate to Hill’s strains through solving the tensor equation AX+XA=C

“…(48) is motivated by the fact that computation of b K is simple, even simpler than computation of K 2 , such that each component b K ij of b K with respect to any coordinate system with orthonormal basis is the sum of two products of the components K ij , while each component of K 2 is the sum of three products of K ij (Ting, 1996). Finally, combination of Eqs.…”

Derivatives of the Stretch, Rotation and Exponential Tensors in n-Dimensional Vector Spaces

“…Let H K¯X/¯T, where X = e Tt . Differentiating the expression X Á ¼ TX with respect to T using equation (20), we get…”