different between eigenvalue vs positone

English

Etymology

eigen- +? value, a partial calque of German Eigenwert.

Pronunciation

• enPR: ??g?n'v?lyo?o, IPA^(key): /?a???n?vælju?/

Noun

eigenvalue (plural eigenvalues)

1. (linear algebra) A scalar, ${\displaystyle \lambda }$, such that there exists a non-zero vector ${\displaystyle x}$ (a corresponding eigenvector) for which the image of ${\displaystyle x}$ under a given linear operator ${\displaystyle \mathrm {A} }$ is equal to the image of ${\displaystyle x}$ under multiplication by ${\displaystyle \lambda }$; i.e. ${\displaystyle \mathrm {A} x=\lambda x}$.
   • 1972, F. V. Atkinson, Multiparameter Eigenvalue Problems, Volume I: Matrices and Compact Operators, Academic Press, page x,

     In the extension, one associates eigenvalues, sets of scalars, with arrays of matrices by considering the singularity of linear combinations of the matrices in the various rows, involving the same coefficients in each case. Attention to this area was called in the early l920's by R. D. Carmichael, who pointed out in addition the enormous variety of mixed eigenvalue problems with several parameters.

   • 2000, Hinne Hettema (translator), J. Von Neumann, E. Wigner, On the Behaviour of Eigenvalues in Adiabatic Processes [1929], Hinne Hettema (editor), Quantum Chemistry: Classic Scientific Papers, World Scientific, page 25,

     For many quantum-mechanical problems it is important to investigate the change of eigenvalues and eigenfunctions with the continuous change of one or more parameters. The case in which one knows the eigenvalues and eigenfunctions for two special values of the parameters, and is interested in the region in between is particularly interesting.

   • 2005, Leonid D. Akulenko, Sergei V. Nesterov, High-Precision Methods in Eigenvalue Problems and Their Applications, CRC Press (Chapman & Hall), page 1,

     Problems that require an investigation of eigenvalues and eigenfunctions arise in connection with numerous topics in mechanics, the theory of vibrations and stability, hydrodynamics, elasticity, acoustics, electrodynamics, quantum mechanics, etc.

Usage notes

When unqualified, as in the above example, eigenvalue conventionally refers to a right eigenvalue, characterised by ${\displaystyle \mathrm {M} x=\lambda x}$ for some right eigenvector ${\displaystyle x}$. Left eigenvalues, characterised by ${\displaystyle y\mathrm {M} =y\lambda }$ also exist with associated left eigenvectors ${\displaystyle y}$. (In consequence of the equations, left eigenvectors are row vectors, while right eigenvectors are column vectors.) The convention of right eigenvector as "standard" is fundamentally an arbitrary choice.

Synonyms

• (scalar multiplier of an eigenvector): characteristic root, characteristic value, eigenroot, latent value, proper value

Translations

See also

• eigenbasis
• eigendecomposition, eigen decomposition
• eigenface
• eigenfunction
• eigenmode
• eigenstate
• eigensystem
• eigenvector

Further reading

• Eigenvalues and eigenvectors on Wikipedia.Wikipedia
• Transformation matrix on Wikipedia.Wikipedia
• Eigenvalue on MathWorld.
• Eigen value on Encyclopedia of Mathematics.

English

Etymology

Blend of positive +? monotone

Adjective

positone (not comparable)

1. (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, page 704

     Finally, we mention that several papers studied nonlinear eigenvalue problems of the form

     ${\displaystyle {\begin{cases}-\Delta x(z)=\lambda f(x(z)){\text{ for a.a. }}z\in \Omega ,\\x|_{\partial \Omega },\ x\geq 0\end{cases}}}$

     for ${\displaystyle \scriptstyle \lambda \ >\ 0}$ under the assumption that ${\displaystyle \scriptstyle f:\ \mathbb {R} \ \to \ \mathbb {R} }$ is continuous, positive, monotone. For this reason such problems were named positone... If the nonlinearity ${\displaystyle \scriptstyle f:\ \mathbb {R} \ \to \ \mathbb {R} }$ is continuous, monotone and ${\displaystyle \scriptstyle f(0)\ <\ 0}$,...then the eigenvalue problem is called semipositone...

Derived terms

• semipositone

Italian

Noun

positone m (plural positoni)

1. Alternative form of positrone