Full moment tensor inversion has become a standard method for understanding the mechanisms of earthquakes as the resolution of the inversion process increases. Thus, it is important to know the possible forms of non–double‐couple (non‐DC) moment tensors, which can be obtained because of either the different source mechanisms or the anisotropy of the focal regions. In this study, the form of the moment tensors of seismic sources occurring in transversely isotropic (TI) focal regions is obtained using the eigendecomposition of the elasticity tensor. More precisely, a moment tensor is obtained as a linear combination of the eigenspaces of TI elasticity tensor in which the coefficients of the terms are the corresponding eigenvalues multiplied with the projection of the potency tensor onto the corresponding eigenspaces. Moreover, the eigendecomposition method is also applied to obtain the three different forms of moment tensors in isotropic focal regions, in particular, for the shear source, tensile source, and for any type of potency tensor whose rank is three. This linear algebra point of view makes the structure of the moment tensors more apparent; for example, a shear source tensor is an eigenvector of isotropic elasticity tensor, and hence the resulting moment tensor is proportional to its shear source tensor. Moreover, a geometric interpretation for the scalar seismic moment, which is the norm of the moment tensor, for anisotropic focal regions is achieved through the eigendecomposition method. This method also gives a simple way to quantify the percentage of the isotropic component of the moment tensor of shear sources in TI focal regions. Hence, the complexities in the moment tensor introduced by the anisotropy of the focal region and by the source mechanism can be differentiated.