# What are the importances of eigen values and eigen vectors?

The reason why eigenvalues are so important in mathematics are too many. Here is a very short and extremely incomplete list of the main applications I encountered in my path and that are coming now in mind to me:

**Theoretical applications:**- The eigenvalues of the Jacobian of a vector field at a given point determines the local geometry of the flow and the stability of that point;
- An iterative method yk+1=Ayk is convergent if the spectral radius ρ(A) (the maximum absolute value of the eigenvalues of A) is < 1.

**Practical applications:****Google Page Rank:**The order in which your search results appear in Google is determined by computing an eigen vector.**Face Recognition:**You can automatically recognize faces by computing eigenvectors of images.- In
**mechanics**, the eigen vectors of the moment of inertia tensor define the principal axes of a rigid body. The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass. **Geology and glaciology:**In geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers.**Molecular orbitals:**In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree–Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator.**Vibration analysis**: Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom.

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