Eigenvalues and eigenvectors

In this shear mapping the red arrow changes direction but the blue arrow does not. Therefore the blue arrow is an eigenvector, with eigenvalue 1 as its length is unchanged.

The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, either remain proportional to the original vector (that is, change only in magnitude, not in direction) or become zero. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector changes when multiplied by the matrix. The prefix eigen- is adopted from the German word "eigen" for "own" in the sense of a characteristic description. The eigenvectors are sometimes also called characteristic vectors. Similarly, the eigenvalues are also known as characteristic values.

Mathematicians would say this as: if A is a square matrix, a non-zero vector v is an eigenvector of A if there is a scalar λ (lambda) such that

$A\mathbf{v} = \lambda \mathbf{v} \, .$

The scalar λ (lambda) is said to be the eigenvalue of A corresponding to v.

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 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

For the matrix A

$A = \begin{bmatrix} 2 & 1\\1 & 2 \end{bmatrix}.$

the vector

$\mathbf x = \begin{bmatrix} 3 \\ -3 \end{bmatrix}$

is an eigenvector with eigenvalue 1. Indeed,

$A \mathbf x = \begin{bmatrix} 2 & 1\\1 & 2 \end{bmatrix} \begin{bmatrix} 3 \\ -3 \end{bmatrix} = \begin{bmatrix} (2 \cdot 3) + (1 \cdot (-3)) \\ (1 \cdot 3) + (2 \cdot (-3)) \end{bmatrix} = \begin{bmatrix} 3 \\ -3 \end{bmatrix} = 1 \cdot \begin{bmatrix} 3 \\ -3 \end{bmatrix}.$

On the other hand the vector

$\mathbf x = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$

is not an eigenvector, since

$\begin{bmatrix} 2 & 1\\1 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} (2 \cdot 0) + (1 \cdot 1) \\ (1 \cdot 0) + (2 \cdot 1) \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}.$

and this vector is not a multiple of the original vector x.

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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

• 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
• Eigenvalues and Eigenvectors on the Ask Dr. Math forums: [1], [2]

Online calculators

• arndt-bruenner.de
• bluebit.gr
• wims.unice.fr