Some Applications of the Eigenvalues and Eigenvectors of a square matrix 1.Communication systems: Eigenvalues 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 waterfilling 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.
2.Designing bridges: The natural frequency of the bridge is the eigenvalue of smallest magnitude of a system that models the bridge. The engineers exploit this knowledge to ensure the stability of their constructions. [Watch the video on the collapse of the Tacoma Narrow Bridge which was built in 1940]
3.Designing car stereo system: Eigenvalue analysis is also used in the design of the car stereo systems, where it helps to reproduce the vibration of the car due to the music.
4.Electrical Engineering: The application of eigenvalues and eigenvectors is useful for decoupling three-phase systems through symmetrical component transformation.
5.Mechanical Engineering: 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 “principle 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.