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Eigenvalue vs Positone - What's the difference?

eigenvalue | positone |

As a noun eigenvalue

is (linear algebra) a scalar, \lambda\!, such that there exists a vector x (the corresponding eigenvector) for which the image of x under a given linear operator \rm a\! is equal to the image of x under multiplication by \lambda; ie {\rm a} x = \lambda x\!.

As an adjective positone is

(mathematics) of a particular kind of eigenvalue problem involving a nonlinear function on the reals that is continuous, positive, and monotone.

eigenvalue

English

Noun

(en noun)
  • (linear algebra) A scalar, \lambda\!, such that there exists a vector x (the corresponding eigenvector) for which the image of x under a given linear operator \rm A\! is equal to the image of x under multiplication by \lambda; i.e. {\rm A} x = \lambda x\!
    • ''The eigenvalues \lambda\! of a square transformation matrix \rm M\! may be found by solving \det({\rm M} - \lambda {\rm I}) = 0\! .

    Usage notes

    When unqualified, as in the above example, eigenvalue conventionally refers to a right eigenvalue, characterised by {\rm M} x = \lambda x\! for some right eigenvector x\!. Left eigenvalues, charactarised by y {\rm M} = y \lambda\! also exist with associated left eigenvectors y\!. For commutative operators, the left eigenvalues and right eigenvalues will be the same, and are referred to as eigenvalues with no qualifier.

    Synonyms

    * characteristic root * characteristic value * eigenroot * latent value * proper value

    Related terms

    * eigen decomposition * eigenface * eigenfunction * eigenmode * eigenstate * eigenvector

    See also

    * ("eigenvalue" on Wikipedia) * Mathworld article on eigenvalues

    positone

    English

    Adjective

    (en adjective)
  • (mathematics) of a particular kind of eigenvalue problem involving a nonlinear function on the reals that is continuous, positive, and monotone.
  • * 2004 , Leszek Gasinski, Nikolaos S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems , CRC Press, 2004 ISBN 1420035037, page 704
    • Finally, we mention that several papers studied nonlinear eigenvalue problems of the form
  • *::
    • \begin{cases} -\Delta x(z) = \lambda f (x (z)) \text { for a.a. }z \in \Omega, \\ x, _{\partial \Omega},\ x \ge 0 \end{cases}
  • *:for \scriptstyle \lambda\ >\ 0 under the assumption that \scriptstyle f:\ \mathbb R\ \to\ \mathbb R is continuous, positive, monotone. For this reason such problems were named positone'' ... If the nonlinearity \scriptstyle f:\ \mathbb R\ \to\ \mathbb R is continuous, monotone and \scriptstyle f(0)\ <\ 0 ,...the the eigenvalue problem is called ''semipositone ...

    • Derived terms

    *semipositone