القيمة الذاتية لنظام ما eigen value أوبالأحرى في معادلة تفاضلية هوعدد يمكن حتىقد يكون من الأعداد السليمة أومن الأعداد المعقدة.

مقاربة الرياضيات التفاضلية للقيمة الذاتية

حساب حل معادلة تفاضلية من الدرجة الأولى:

فلنعتبر مثلا المعادلة التفاضلية البسيطة التالية(مع غض الطرف مبدئيا عن وجوب إعتبار الشروط الأولى أي initial conditions عند حساب الحل):

${\displaystyle {\dot {x +10x=0$

ولنحاول البحث عن حل هذه المعادلة. المعروف هوأنه يمكن حتى نقول حتى حل هذالمعادلة هو:

${\displaystyle e^{ct$

أي حتى ${\displaystyle x=e^{ct$ وإذا عوضت x ب ${\displaystyle e^{ct$ فإنك تحصل على المعادلة التالية:

${\displaystyle {\dot {(e^{ct ) +10e^{ct =0$

أي ${\displaystyle ce^{ct +10e^{ct =0$

أي بعد حتى نشطب ${\displaystyle e^{ct$ من المعادلة فإنك تحصل على المعادلة ${\displaystyle c+10=0$ وهذا بدوره يعني حتى c=-10.

القيمة الذاتية

في عملية الحساب أعلاه وصلنا من معادلة تفاضلية من الدرجة الأولى إلى معادلة بسيطة لحساب القيمة الذاتية c ألا وهي المعادلة ${\displaystyle -c+10=0$ ويمكن تعميم هذه الطريقة أي كيفية الوصول من معادلة تفاضلية ذات درجة 2 أوثلاثة إلخ...إلى المعادلة لحساب القيمة الذاتية أوفي هذه الحالة القيم الذاتية (لأن درجة العلاقة التفاضلية تتطابق دائما مع عدد القيم الخاصة التي تحسبها حيثقد يكون عليك في هذه الحالة حل بولينومات polynoms وحتى تراعي طبعا حتى بعض الحلول قد تكون مكررة أي أنه يجب حتى تعدها عدة مرات حتى تكون ملاحظتي هذه سليمة).
كما يجدر الإشارة إلى حتى هذه الطريقة أوالمعالجة حيث فقط للنظم أوالمعادلات التفاضلية الخطية أي أنه في صورة إنعدام الخطية لا يمكن الحديث عن قيمة ذاتية.
كما حتى القيمة الذاتية تفهمنا إذا كان نظام ما مستقرا (إذا كانت القيمة الذاتية سالبة) أوغير مستقر (إذا كانت القيمة موجبة). وهي كذلك مرشد على سرعة النظام أوسرعة رده (إذا كانت القيمة المطلقة للقيمة الذاتية كبيرة فإن النظام سريع أي سرعة ردة عمله سريعة).

مقاربة الرياضيات الخطية للقيمة الذاتية

التطبيقات

القيم الذاتية للتحويلات الهندسية

The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors.

	Scaling	Unequal scaling	Rotation	Horizontal shear	Hyperbolic rotation
Illustration
Matrix	${\displaystyle {\begin{bmatrix k&0\\0&k\end{bmatrix$	${\displaystyle {\begin{bmatrix k_{1 &0\\0&k_{2 \end{bmatrix$	${\displaystyle {\begin{bmatrix c&-s\\s&c\end{bmatrix$ ${\displaystyle c=\cos \theta$ ${\displaystyle s=\sin \theta$	${\displaystyle {\begin{bmatrix 1&k\\0&1\end{bmatrix$	${\displaystyle {\begin{bmatrix c&s\\s&c\end{bmatrix$ ${\displaystyle c=\cosh \varphi$ ${\displaystyle s=\sinh \varphi$
Characteristic polynomial	${\displaystyle \ (\lambda -k)^{2$	${\displaystyle (\lambda -k_{1 )(\lambda -k_{2 )$	${\displaystyle \lambda ^{2 -2c\lambda +1$	${\displaystyle \ (\lambda -1)^{2$	${\displaystyle \lambda ^{2 -2c\lambda +1$
Eigenvalues, ${\displaystyle \lambda _{i$	${\displaystyle \lambda _{1 =\lambda _{2 =k$	${\displaystyle \lambda _{1 =k_{1$ ${\displaystyle \lambda _{2 =k_{2$	${\displaystyle \lambda _{1 =e^{\mathbf {i \theta =c+s\mathbf {i$ ${\displaystyle \lambda _{2 =e^{-\mathbf {i \theta =c-s\mathbf {i$	${\displaystyle \lambda _{1 =\lambda _{2 =1$	${\displaystyle \lambda _{1 =e^{\varphi$ ${\displaystyle \lambda _{2 =e^{-\varphi$,
Algebraic mult., ${\displaystyle \mu _{i =\mu (\lambda _{i )$	${\displaystyle \mu _{1 =2$	${\displaystyle \mu _{1 =1$ ${\displaystyle \mu _{2 =1$	${\displaystyle \mu _{1 =1$ ${\displaystyle \mu _{2 =1$	${\displaystyle \mu _{1 =2$	${\displaystyle \mu _{1 =1$ ${\displaystyle \mu _{2 =1$
Geometric mult., ${\displaystyle \gamma _{i =\gamma (\lambda _{i )$	${\displaystyle \gamma _{1 =2$	${\displaystyle \gamma _{1 =1$ ${\displaystyle \gamma _{2 =1$	${\displaystyle \gamma _{1 =1$ ${\displaystyle \gamma _{2 =1$	${\displaystyle \gamma _{1 =1$	${\displaystyle \gamma _{1 =1$ ${\displaystyle \gamma _{2 =1$
Eigenvectors	All nonzero vectors	${\displaystyle {\begin{aligned u_{1 &={\begin{bmatrix 1\\0\end{bmatrix \\u_{2 &={\begin{bmatrix 0\\1\end{bmatrix \end{aligned$	${\displaystyle {\begin{aligned u_{1 &={\begin{bmatrix {\ 1\\-\mathbf {i \end{bmatrix \\u_{2 &={\begin{bmatrix {\ 1\\+\mathbf {i \end{bmatrix \end{aligned$	${\displaystyle u_{1 ={\begin{bmatrix 1\\0\end{bmatrix$	${\displaystyle {\begin{aligned u_{1 &={\begin{bmatrix {\ 1\\{\ 1\end{bmatrix \\u_{2 &={\begin{bmatrix {\ 1\\-1\end{bmatrix .\end{aligned$

The characteristic equation for a rotation is a quadratic equation with discriminant ${\displaystyle D=-4(\sin \theta )^{2$, which is a negative number whenever θ is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, ${\displaystyle \cos \theta \pm \mathbf {i \sin \theta$; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane.

A linear transformation that takes a square to a rectangle of the same area (a squeeze mapping) has reciprocal eigenvalues.

معادلة شرودنگر

An example of an eigenvalue equation where the transformation ${\displaystyle T}$

${\displaystyle H\psi _{E =E\psi _{E \,$

where ${\displaystyle H}$${\displaystyle {\mathrm{\psi <!--\; \psi \; -->}}_E}$${\displaystyle E}$

Principal component analysis

Vibration analysis

Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by

${\displaystyle m{\ddot {x +kx=0$

or

${\displaystyle m{\ddot {x =-kx$

that is, acceleration is proportional to position (i.e., we expect ${\displaystyle x$ to be sinusoidal in time).

In ${\displaystyle n}$${\displaystyle k}$

${\displaystyle kx=\omega ^{2 mx$

where ${\displaystyle {\mathrm{\omega <!--\; \omega \; -->}}^2}$${\displaystyle k}$

${\displaystyle m{\ddot {x +c{\dot {x +kx=0$

leads to a so-called quadratic eigenvalue problem,

${\displaystyle \left(\omega ^{2 m+\omega c+k\right)x=0.$

This can be reduced to a generalized eigenvalue problem by algebraic manipulation at the cost of solving a larger system.

The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using finite element analysis, but neatly generalize the solution to scalar-valued vibration problems.

Eigenfaces

In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel. The dimension of this vector space is the number of pixels. The eigenvectors of the covariance matrix associated with a large set of normalized pictures of faces are called eigenfaces; this is an example of principal component analysis. They are very useful for expressing any face image as a linear combination of some of them. In the facial recognition branch of biometrics, eigenfaces provide a means of applying data compression to faces for identification purposes. Research related to eigen vision systems determining hand gestures has also been made.

Similar to this concept, eigenvoices represent the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech recognition systems for speaker adaptation.

See also

• Antieigenvalue theory
• Eigenoperator
• Eigenplane
• Eigenvalue algorithm
• Introduction to eigenstates
• Jordan normal form
• List of numerical analysis software
• Nonlinear eigenproblem
• Quadratic eigenvalue problem
• Singular value

Notes

References

Citations

Sources

External links

الوصلات الخارجية في هذه الموضوعة قد تكون غير متبعة لسياسات المحتوى في الفهم أوتعليماتها. فضلاً طور هذه الموضوعة بإزالة الوصلات الخارجية غير المناسبة أوالزائدة. (December 2019)

• What are Eigen Values? – non-technical introduction from PhysLink.com's "Ask the Experts"
• Eigen Values and Eigen Vectors Numerical Examples – Tutorial and Interactive Program from Revoledu.
• Introduction to Eigen Vectors and Eigen Values – lecture from Khan Academy
• Hill, Roger (2009). "λ – Eigenvalues". Sixty Symbols. Brady Haran for the University of Nottingham.
• "A Beginner's Guide to Eigenvectors". Deeplearning4j. 2015. Archived from the original on 2018-07-21. Retrieved 2015-08-18.
• Eigenvectors and eigenvalues | Essence of linear algebra, chapterعشرة – A visual explanation with 3Blue1Brown
• Matrix Eigenvectors Calculator from Symbolab (Click on the bottom right button of the 2x12 grid to select a matrix size. Select an ${\displaystyle n\times n$ size (for a square matrix), then fill out the entries numerically and click on the Go button. It can accept complex numbers as well.)

Theory

• Hazewinkel, Michiel, ed. (2001), "Eigen value", Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Hazewinkel, Michiel, ed. (2001), "Eigen vector", Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Eigenvalue (of a matrix) على بلانيت ماث
• Eigenvector – Wolfram MathWorld
• Eigen Vector Examination working applet
• Same Eigen Vector Examination as above in a Flash demo with sound
• Computation of Eigenvalues
• Numerical solution of eigenvalue problems Edited by Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk van der Vorst
• Eigenvalues and Eigenvectors on the Ask Dr. Math forums: [1], [2]

Demonstration applets

• Java applet about eigenvectors in the real plane
• Wolfram Language functionality for Eigenvalues, Eigenvectors and Eigensystems