Eigenvalue

An Eigenvalue is a real number, [math]\displaystyle{ \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]\displaystyle{ V }[/math] be a vector space over the field [math]\displaystyle{ F }[/math] and let [math]\displaystyle{ T }[/math] be a linear operator on [math]\displaystyle{ V }[/math]. An eigenvalue of [math]\displaystyle{ T }[/math] is a scalar [math]\displaystyle{ \lambda }[/math] in [math]\displaystyle{ F }[/math] such that there is a non-zero vector [math]\displaystyle{ X }[/math] in [math]\displaystyle{ V }[/math] with [math]\displaystyle{ TX=\lambda X }[/math].Here [math]\displaystyle{ X }[/math] is called the eigen vector of T for respective eigen value [math]\displaystyle{ \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]\displaystyle{ \lambda }[/math] is an eigenvalue of a matrix [math]\displaystyle{ A }[/math] and [math]\displaystyle{ \alpha }[/math] be any scalar then
    • The matrix [math]\displaystyle{ \alpha A }[/math] has eigenvalue [math]\displaystyle{ \alpha \lambda }[/math].
    • The matrix [math]\displaystyle{ A^m }[/math] has eigenvalue [math]\displaystyle{ \lambda^m }[/math].
    • The matrix [math]\displaystyle{ A-kI }[/math] has eigenvalue [math]\displaystyle{ \lambda-k }[/math].
    • The matrix [math]\displaystyle{ A^{-1} }[/math] has eigenvalue [math]\displaystyle{ \frac{1}{\lambda} }[/math].
    • The matrix [math]\displaystyle{ A }[/math] and [math]\displaystyle{ A^T }[/math] have same eigenvalues.
• Example(s):
  • [math]\displaystyle{ \operatorname{Eigenvalues} \Bigl(\begin{bmatrix} 5 & 4 \\ 1 & 2 \end{bmatrix} \Bigr) = \{6,1\} }[/math].
  • [math]\displaystyle{ \operatorname{Eigenvalues} \Bigl(\begin{bmatrix} 1 & 2 \\ -2 & 1 \end{bmatrix} \Bigr) = \{ 1+2i , 1-2i \} }[/math].
  • [math]\displaystyle{ \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.

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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]\displaystyle{ 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.

2013

• http://en.wikipedia.org/wiki/Eigenvalue
  • An eigenvector of a square matrix [math]\displaystyle{ A }[/math] is a non-zero vector [math]\displaystyle{ v }[/math] that, when multiplied by [math]\displaystyle{ A }[/math], yields the original vector multiplied by a single number [math]\displaystyle{ \lambda }[/math]; that is: [math]\displaystyle{ A v = \lambda v }[/math] The number [math]\displaystyle{ \lambda }[/math] is called the eigenvalue of [math]\displaystyle{ A }[/math] corresponding to [math]\displaystyle{ v }[/math]. … Thus, for example, the exponential function [math]\displaystyle{ f(x) = a^x }[/math] is an eigenfunction of the derivative operator " [math]\displaystyle{ {}' }[/math] ", with eigenvalue [math]\displaystyle{ \lambda = \ln a }[/math], since its derivative is [math]\displaystyle{ 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]\displaystyle{ A }[/math] is the set of all eigenvectors with the same eigenvalue, together with the zero vector. An eigenbasis for [math]\displaystyle{ A }[/math] is any basis for the set of all vectors that consists of linearly independent eigenvectors of [math]\displaystyle{ A }[/math]. Not every real matrix has real eigenvalues, but every complex matrix has at least one complex eigenvalue.

2012

• http://mathworld.wolfram.com/Eigenvalue.html
  • Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144).

2010

• (WordNet, 2010) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=eigenvalue
  • S: (n) eigenvalue, eigenvalue of a matrix, eigenvalue of a square matrix, characteristic root of a square matrix ((mathematics) any number such that a given square matrix minus that number times the identity matrix has a zero determinant)

2009

• (Gentle, 2009) ⇒ James E. Gentle. (2009). “Computational Statistics." Springer. ISBN:978-0-387-98143-7
  • QUOTE: Multiplication of a given vector by a square matrix may result in a scalar multiple of the vector. If [math]\displaystyle{ A }[/math] is a [math]\displaystyle{ n \times n }[/math] matrix, [math]\displaystyle{ v }[/math] is a vector not equal to [math]\displaystyle{ 0 }[/math], and [math]\displaystyle{ c }[/math] is a scalar such that [math]\displaystyle{ Av = cv, }[/math] we say [math]\displaystyle{ v }[/math] is an eigenvector of [math]\displaystyle{ A }[/math] and [math]\displaystyle{ c }[/math] is an eigenvalue of [math]\displaystyle{ A }[/math].

    … The effect of a matrix multiplication of an eigenvector is the same as a scalar multiplication of the eigenvector. The eigenvector is an invariant of the transformation in the sense that its direction does not change under the matrix multiplication transformation. Thies would seem to indicate that the eigenvector and eigenvalue depend on some kind of deep properties of the matrix, and indeed, this is the case.

    We immmediately see that if an eigenvalue of a matrix [math]\displaystyle{ A }[/math] is [math]\displaystyle{ 0 }[/math], then [math]\displaystyle{ A }[/math] must be singular.

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