What is the importance of eigenvalues/eigenvectors? What is the importance of eigenvalues/eigenvectors?  

When you apply transformations to the systems/objects represented by matrices, and you need some characteristics of these matrices you have to calculate eigenvectors (eigenvalues). 

"Having an eigenvalue is an accidental property of a real matrix (since it may fail to have an eigenvalue), but every complex matrix has an eigenvalue."(Wikipedia) 

Eigenvalues ​​characterize important properties of linear transformations, such as whether a system of linear equations has a unique solution or not. In many applications eigenvalues ​​also describe physical properties of a mathematical model.
Some important applications -


*

*Principal Components Analysis (PCA) in object/image recognition;  

*Physics - stability analysis, the physics of rotating bodies;

*Market risk analysis - to define if a matrix is positive definite;

*PageRank from Google.
