.. _eig:

Application of Eigenvalues and Eigenvectors
===========================================

The eigenvalues and eigenvectors of a matrix have several applications
in science, engineering, and mathematics. Eigenvalues and eigenvectors
are used to solve systems of difference and differential equations, but
more generally, they are used for data analysis, where the matrix
:math:`\bf{A}` represents data rather than coefficients of a system of
equations. They introduce a simple, yet powerful relationship between a
matrix and a set of special vectors and scalar values, which provides
elegant solutions to some otherwise difficult problems.

Eigenvalues and eigenvectors are closely related to the :ref:`SVD`, which
has many data analysis applications and is used by numerical software in
many linear algebra functions that we used in :ref:`linAlg` to solve systems
of equations.  Eigenvalues and eigenvectors are also used in :ref:`PCA`, a
data analysis tool for classification and recognition applications.

.. index:: eigenvalue
.. index:: eigenvector

**Contents**

.. toctree::
   :maxdepth: 2

   eigIntro
   eigDemo
   find_eig
   eigProperties
   powers
   chBasis