Eigenvalues and eigenvectors

Linear algebra talks about types of functions called _transformations_. In that context, an **eigenvector** is a vector—different from the null vector—which does not change direction after the transformation (except if the transformation turns the vector to the opposite direction). The vector may change its length, or become zero ("null"). The **eigenvalue** is the value of the vector's change in length, and is typically denoted by the symbol . The word "eigen" is a German word, which means "own" or "typical".

## Basics

If there exists a **square matrix** called ***A***, a **scalar _λ***, and a **non-zero vector** **v**, then ***λ**_ **is the eigenvalue** and **v is the eigenvector** if the following equation is satisfied:

In other words, if matrix _A_ times the vector **v** is equal to the scalar _λ_ times the vector **v**, then _λ_ is the eigenvalue of **v**, where **v** is the eigenvector.

An **eigenspace** of _A_ is the set of all eigenvectors with the same eigenvalue together with the zero vector. However, the zero vector is not an eigenvector.

These ideas often are extended to more general situations, where scalars are elements of any field, vectors are elements of any vector space, and linear transformations may or may not be represented by matrix multiplication. For example, instead of real numbers, scalars may be complex numbers; instead of arrows, vectors may be functions or frequencies; instead of matrix multiplication, linear transformations may be operators such as the derivative from calculus. These are only a few of countless examples where eigenvectors and eigenvalues are important.

In cases like these, the idea of _direction_ loses its ordinary meaning, and has a more abstract definition instead. But even in this case, if that abstract _direction_ is unchanged by a given linear transformation, the prefix "eigen" is used, as in _eigenfunction_, _eigenmode_, _eigenface_, _eigenstate_, and _eigenfrequency_.

Eigenvalues and eigenvectors have many applications in both pure and applied mathematics. They are used in matrix factorization, quantum mechanics, facial recognition systems, and many other areas.

## Example

For the matrix _A_
the vector
is an eigenvector with eigenvalue 1. Indeed, one can verify that:
On the other hand the vector
is _not_ an eigenvector, since
and this vector is not a multiple of the original vector **x**.

## Related pages

- Matrix analysis
- Numerical linear algebra
- Singular value decomposition
- Wikiversity:Eigenvalues for beginners

## References

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- Pigolkina, T. S. and Shulman, V. S., _Eigenvalue_ (in Russian), In:Vinogradov, I. M. (Ed.), _Mathematical Encyclopedia_, Vol. 5, Soviet Encyclopedia, Moscow, 1977.
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- Curtis, Charles W., _Linear Algebra: An Introductory Approach_, 347 p., Springer; 4th ed. 1984. Corr. 7th printing edition (August 19, 1999), .
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## Other websites

- What are Eigen Values? — non-technical introduction from PhysLink.com's "Ask the Experts"
- Introduction to Eigen Vectors and Eigen Values  – lecture from Kahn Academy

- Theory
- Eigen Vector Examination working applet
- Same Eigen Vector Examination as above in a Flash demo with sound
- Computation of Eigenvalues
- Numerical solution of eigenvalue problems, edited by Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe and Henk van der Vorst

- Online calculators
- arndt-bruenner.de
- bluebit.gr
- wims.unice.fr