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In this paper, we discuss the generalized quaternion matrix equation AXB+CX⋆D=E, where X⋆ is one of X, X*, the η‐conjugate or the η‐conjugate transpose of X with η∈{i,j,k}. Two new real representations of a generalized quaternion matrix are proposed. By using this method, the criteria for the existence and uniqueness of solutions to the mentioned matrix equation as well as the existence of X=± X⋆ solutions to the generalized quaternion matrix equation AXB+CXD=E are derived in a unified way.

We construct microlocal solutions of Rayleigh and Stoneley waves in isotropic linear elasticity with the density and the Lamé parameters smooth up to a curved boundary or interface. We compute the direction of the microlocal polarization and show a retrograde elliptical motion of these two type of waves.

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