The application of eigenvalues and eigenvectors is useful for decoupling three-phase systems through symmetrical component transformation.
Eigenvalues and eigenvectors allow us to “reduce” a linear operation to separate, simpler, problems. For example, if a stress is applied to a “plastic” solid, the deformation can be dissected into “principal directions”- those directions in which the deformation is greatest. Vectors in the principle directions are the eigenvectors and the percentage deformation in each principle direction is the corresponding eigenvalue.
Eigen values were used by Claude Shannon to determine the theoretical limit to how much information can be transmitted through a communication medium like your telephone line or through the air. This is done by calculating the eigenvectors and eigenvalues of the communication channel (expressed a matrix), and then water filling on the eigenvalues. The eigenvalues are then, in essence, the gains of the fundamental modes of the channel, which themselves are captured by the eigenvectors.