Eigenvalue

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An Eigenvalue is a real number, [math]\lambda[/math] that when multiplied with an eigenvector yields the original vector.

  • AKA: Characteristic Root, Latent Value, Proper Value, Spectral Value.
  • Context:
    • It can be produced by an Eigenvalue Finding Task.
    • It be represented as:
      Let [math]V[/math] be a vector space over the field [math]F[/math] and let [math]T[/math] be a linear operator on [math]V[/math]. An eigenvalue of [math]T[/math] is a scalar [math]\lambda[/math] in [math]F[/math] such that there is a non-zero vector [math]X[/math] in [math]V[/math] with [math]TX=\lambda X[/math].Here [math]X[/math] is called the eigen vector of T for respective eigen value [math]\lambda[/math].
    • It can used in a Dimension-Compression Task (along with its eigenvectors).
    • When a beam is struck, its natural frequency ( eigenvalues) can be measured. So eigenvalues can be used to test for cracks and deformities in the structural components used for construction.
    • If [math]\lambda[/math] is an eigenvalue of a matrix [math]A[/math] and [math]\alpha[/math] be any scalar then
      • The matrix [math]\alpha A[/math] has eigenvalue [math]\alpha \lambda[/math].
      • The matrix [math]A^m[/math] has eigenvalue [math] \lambda^m[/math].
      • The matrix [math]A-kI[/math] has eigenvalue [math]\lambda-k[/math].
      • The matrix [math]A^{-1}[/math] has eigenvalue [math]\frac{1}{\lambda}[/math].
      • The matrix [math]A[/math] and [math]A^T[/math] have same eigenvalues.
  • Example(s):
    • [math]\operatorname{Eigenvalues} \Bigl(\begin{bmatrix} 5 & 4 \\ 1 & 2 \end{bmatrix} \Bigr) = \{6,1\}[/math].
    • [math]\operatorname{Eigenvalues} \Bigl(\begin{bmatrix} 1 & 2 \\ -2 & 1 \end{bmatrix} \Bigr) = \{ 1+2i , 1-2i \}[/math].
    • [math]\operatorname{Eigenvalues} \Bigl(\begin{bmatrix} -2 & 2 & 3 \\ 2 & 1 & -6 \\ -1 & -2 & 0 \end{bmatrix}\Bigr) = \{ 5, -3 , -3 \}[/math] .
  • See: Matrix Equation, Square Matrix.


References

2015

  • (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors Retrieved:2015-2-16.
    • An eigenvector or characteristic vector of a linear transformation defines a direction that is invariant under the transformation. Let the transformation be defined by the square matrix A, then an invariant direction of A is the non-zero vector v, that has the property that the product Av is a scalar multiple of v. This is written as the equation, : [math]A\mathbf{v} = \lambda \mathbf{v},[/math] where λ is known as the eigenvalue associated with the eigenvector v.

      (Because this equation uses post-multiplication of the matrix A by the vector v it describes an right eigenvector.) The number λ is called the eigenvalue or characteristic value of A corresponding to v.[1]

  1. Wolfram Research, Inc. (2010) Eigenvector. Accessed on 2010-01-29.

2013

  • http://en.wikipedia.org/wiki/Eigenvalue
    • An eigenvector of a square matrix [math]A[/math] is a non-zero vector [math]v[/math] that, when multiplied by [math]A[/math], yields the original vector multiplied by a single number [math]\lambda[/math]; that is: [math]A v = \lambda v[/math] The number [math]\lambda[/math] is called the eigenvalue of [math]A[/math] corresponding to [math]v[/math]. … Thus, for example, the exponential function [math]f(x) = a^x[/math] is an eigenfunction of the derivative operator " [math]{}'[/math] ", with eigenvalue [math]\lambda = \ln a[/math], since its derivative is [math]f'(x) = (\ln a)a^x = \lambda f(x)[/math].

      The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding eigenvalue, is called the eigensystem of that matrix. An eigenspace of a matrix [math]A[/math] is the set of all eigenvectors with the same eigenvalue, together with the zero vector. An eigenbasis for [math]A[/math] is any basis for the set of all vectors that consists of linearly independent eigenvectors of [math]A[/math]. Not every real matrix has real eigenvalues, but every complex matrix has at least one complex eigenvalue.

2012

2010

2009