Significance of eigenvalues and eigenvectors in control system

The eigenvalues and eigenvectors of the system determine the relationship between the individual system state variables (the members of the x vector), the response of the system to inputs, and the stability of the system.

1. What is the significance of eigenvalues and eigenvectors?
2. How eigenvalues and eigenvectors are related to the stability of dynamic systems?
3. What is the physical significance of eigenvalues?
4. How eigen values are used in communication system?

What is the significance of eigenvalues and eigenvectors?

Eigenvalues are the special set of scalar values that is associated with the set of linear equations most probably in the matrix equations. The eigenvectors are also termed as characteristic roots. It is a non-zero vector that can be changed at most by its scalar factor after the application of linear transformations.

How eigenvalues and eigenvectors are related to the stability of dynamic systems?

If the set of eigenvalues for the system has repeated real eigenvalues, then the stability of the critical point depends on whether the eigenvectors associated with the eigenvalues are linearly independent, or orthogonal. This is the case of degeneracy, where more than one eigenvector is associated with an eigenvalue.

What is the physical significance of eigenvalues?

Eigenvectors make understanding linear transformations easy. They are the "axes" (directions) along which a linear transformation acts simply by "stretching/compressing" and/or "flipping"; eigenvalues give you the factors by which this compression occurs.

How eigen values are used in communication system?

Eigenvalues are used in communication systems to determine the theoretical limit to how much information can be transmitted through a communication medium like your telephone line or through the air.