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Eigenvalues and eigenvectors

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Illustration of a transformation (of Mona Lisa): The image is changed in such a way that the red arrow (vector) does not change its direction, but the blue one does. The red vector therefore is an eigenvector of this transformation, the blue one is not. Since the red vector does not change its length, its eigenvalue is 1. The transformation used is called shear mapping.

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 [math]\displaystyle{ \lambda }[/math].[1] The word "eigen" is a German word, which means "own" or "typical".[2]

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:

[math]\displaystyle{ A\mathbf{v} = \lambda \mathbf{v} \, . }[/math][3]

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.[4]

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

[math]\displaystyle{ A = \begin{bmatrix} 2 & 1\\1 & 2 \end{bmatrix}. }[/math]

the vector

[math]\displaystyle{ \mathbf x = \begin{bmatrix} 3 \\ -3 \end{bmatrix} }[/math]

is an eigenvector with eigenvalue 1. Indeed, one can verify that:

[math]\displaystyle{ 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}. }[/math]

On the other hand the vector

[math]\displaystyle{ \mathbf x = \begin{bmatrix} 0 \\ 1 \end{bmatrix} }[/math]

is not an eigenvector, since

[math]\displaystyle{ \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}. }[/math]

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

Eigenvalues And Eigenvectors Media

  • In this shear mapping the red arrow changes direction, but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping because it does not change direction, and since its length is unchanged, its eigenvalue is 1.

  • A 2×2 real and symmetric matrix representing a stretching and shearing of the plane. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them.

  • Matrix A acts by stretching the vector x, not changing its direction, so x is an eigenvector of A.

  • The transformation matrix [

  • Homothetic stretch of a surface

  • Unit square shrunk in the direction of y and stretched at the direction of x

  • simple graphics describing a rotation around the source of a coordinate system

  • Horizontal shear transformation (mapping)

  • Illustration of a squeeze mapping with parameter r = 1.5.

Related pages

Notes

  1. ↑ "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-19.
  2. ↑ "Eigenvector and Eigenvalue". www.mathsisfun.com. Retrieved 2020-08-19.
  3. ↑ Weisstein, Eric W. "Eigenvalue". mathworld.wolfram.com. Retrieved 2020-08-19.
  4. ↑ Weisstein, Eric W. "Eigenvector". mathworld.wolfram.com. Retrieved 2020-08-19.

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