Eigenvalues and eigenvectors
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Linear algebra talks about functions, which are often called transformations. In that context, an eigenvector is a vector -- different from the null vector -- which does not change direction in the transformation (except if it turns the vector exactly around). The vector may change its length, or become null. The value of the change in length of the vector is known as eigenvalue.
Basics[change | change source]
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.
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 such cases, the concept of direction loses its ordinary meaning, and is given an abstract definition. Even so, 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, in quantum mechanics, facial recognition systems, and in many other areas.
Example[change | change source]
For the matrix A
is an eigenvector with eigenvalue 1. Indeed,
On the other hand the vector
is not an eigenvector, since
and this vector is not a multiple of the original vector x.
Notes[change | change source]
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Other websites[change | change source]
|The English Wikibook Linear Algebra has more information on:|
|The English Wikibook The Book of Mathematical Proofs has more information on:|
- 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
- Eigenvector — Wolfram MathWorld
- 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