قيم ومتجهات ذاتية

القيمة الذاتية لنظام ما eigen value أو بالأحرى في معادلة تفاضلية هو عدد يمكن أن يكون من الأعداد الصحيحة أو من الأعداد المعقدة.

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 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):

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

و لنحاول البحث عن حل هذه المعادلة. المعروف هو أنه يمكن أن نقول أن حل هذالمعادلة هو:

$e^{ct}$

أي أن $x=e^{ct}$ و إذا عوضت x ب $e^{ct}$ فإنك تحصل على المعادلة التالية:

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

أي $ce^{ct}+10e^{ct}=0$

أي بعد أن نشطب $e^{ct}$ من المعادلة فإنك تحصل على المعادلة $c+10=0$ و هذا بدوره يعني أن 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 $-c+10=0$ و يمكن تعميم هذه الطريقة أي كيفية الوصول من معادلة تفاضلية ذات درجة 2 أو 3 إلخ...إلى المعادلة لحساب القيمة الذاتية أو في هذه الحالة القيم الذاتية (لأن درجة العلاقة التفاضلية تتطابق دائما مع عدد القيم الخاصة التي تحسبها حيث يكون عليك في هذه الحالة حل بولينومات polynoms و أن تراعي طبعا أن بعض الحلول قد تكون مكررة أي أنه يجب أن تعدها عدة مرات حتى تكون ملاحظتي هذه صحيحة).
كما يجدر الإشارة إلى أن هذه الطريقة أو المعالجة حيث فقط للنظم أو المعادلات التفاضلية الخطية أي أنه في صورة إنعدام الخطية لا يمكن الحديث عن قيمة ذاتية.
كما أن القيمة الذاتية تعلمنا إذا كان نظام ما مستقرا (إذا كانت القيمة الذاتية سالبة) أو غير مستقر (إذا كانت القيمة موجبة). وهي كذلك دليل على سرعة النظام أو سرعة رده (إذا كانت القيمة المطلقة للقيمة الذاتية كبيرة فإن النظام سريع أي سرعة ردة فعله سريعة).

 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	${\begin{bmatrix}k&0\\0&k\end{bmatrix}}$	${\begin{bmatrix}k_{1}&0\\0&k_{2}\end{bmatrix}}$	${\begin{bmatrix}c&-s\\s&c\end{bmatrix}}$ $c=\cos \theta$ $s=\sin \theta$	${\begin{bmatrix}1&k\\0&1\end{bmatrix}}$	${\begin{bmatrix}c&s\\s&c\end{bmatrix}}$ $c=\cosh \varphi$ $s=\sinh \varphi$
Characteristic polynomial	$\ (\lambda -k)^{2}$	$(\lambda -k_{1})(\lambda -k_{2})$	$\lambda ^{2}-2c\lambda +1$	$\ (\lambda -1)^{2}$	$\lambda ^{2}-2c\lambda +1$
Eigenvalues, $\lambda _{i}$	$\lambda _{1}=\lambda _{2}=k$	$\lambda _{1}=k_{1}$ $\lambda _{2}=k_{2}$	$\lambda _{1}=e^{\mathbf {i} \theta }=c+s\mathbf {i}$ $\lambda _{2}=e^{-\mathbf {i} \theta }=c-s\mathbf {i}$	$\lambda _{1}=\lambda _{2}=1$	$\lambda _{1}=e^{\varphi }$ $\lambda _{2}=e^{-\varphi }$,
Algebraic mult., $\mu _{i}=\mu (\lambda _{i})$	$\mu _{1}=2$	$\mu _{1}=1$ $\mu _{2}=1$	$\mu _{1}=1$ $\mu _{2}=1$	$\mu _{1}=2$	$\mu _{1}=1$ $\mu _{2}=1$
Geometric mult., $\gamma _{i}=\gamma (\lambda _{i})$	$\gamma _{1}=2$	$\gamma _{1}=1$ $\gamma _{2}=1$	$\gamma _{1}=1$ $\gamma _{2}=1$	$\gamma _{1}=1$	$\gamma _{1}=1$ $\gamma _{2}=1$
Eigenvectors	All nonzero vectors	${\begin{aligned}u_{1}&={\begin{bmatrix}1\\0\end{bmatrix}}\\u_{2}&={\begin{bmatrix}0\\1\end{bmatrix}}\end{aligned}}$	${\begin{aligned}u_{1}&={\begin{bmatrix}{\ }1\\-\mathbf {i} \end{bmatrix}}\\u_{2}&={\begin{bmatrix}{\ }1\\+\mathbf {i} \end{bmatrix}}\end{aligned}}$	$u_{1}={\begin{bmatrix}1\\0\end{bmatrix}}$	${\begin{aligned}u_{1}&={\begin{bmatrix}{\ }1\\{\ }1\end{bmatrix}}\\u_{2}&={\begin{bmatrix}{\ }1\\-1\end{bmatrix}}.\end{aligned}}$

The characteristic equation for a rotation is a quadratic equation with discriminant $D=-4(\sin \theta )^{2}$, 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, $\cos \theta \pm \mathbf {i} \sin \theta$; 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

The wavefunctions associated with the bound states of an electron in a hydrogen atom can be seen as the eigenvectors of the hydrogen atom Hamiltonian as well as of the angular momentum operator. They are associated with eigenvalues interpreted as their energies (increasing downward: $n=1,\,2,\,3,\,\ldots$) and angular momentum (increasing across: s, p, d, ...). The illustration shows the square of the absolute value of the wavefunctions. Brighter areas correspond to higher probability density for a position measurement. The center of each figure is the atomic nucleus, a proton.

An example of an eigenvalue equation where the transformation $T$ is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics:

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

where $H$, the Hamiltonian, is a second-order differential operator and $\psi _{E}$, the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue $E$, interpreted as its energy.

 Principal component analysis

PCA of the multivariate Gaussian distribution centered at $(1,3)$ with a standard deviation of 3 in roughly the $(0.878,0.478)$ direction and of 1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite) covariance matrix scaled by the square root of the corresponding eigenvalue. (Just as in the one-dimensional case, the square root is taken because the standard deviation is more readily visualized than the variance.

المقالة الرئيسية: Principal component analysis

انظر أيضاً: Positive semidefinite matrix and Factor analysis

 Vibration analysis

Mode shape of a tuning fork at eigenfrequency 440.09 Hz

المقالة الرئيسية: Vibration

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

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

or

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

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

In $n$ dimensions, $m$ becomes a mass matrix and $k$ a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem

$kx=\omega ^{2}mx$

where $\omega ^{2}$ is the eigenvalue and $\omega$ is the (imaginary) angular frequency. The principal vibration modes are different from the principal compliance modes, which are the eigenvectors of $k$ alone. Furthermore, damped vibration, governed by

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

leads to a so-called quadratic eigenvalue problem,

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

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.

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 Eigenfaces

Eigenfaces as examples of eigenvectors

المقالة الرئيسية: Eigenface

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

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 External links

• 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, chapter 10 – 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 $n\times n$ 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