eigen value, eigen vector and its application

TERM PAPER

ENGINEERING MATHEMATICS (MTH101) Topic:

EIGEN EIGEN VALUES VALUES AND AND EIGE EIGEN N VECTORS AND ITS APPLICATIONS

DOA: 14 Sep 2010 DOR: 19 Oct 2010 DOS: 16 Nov 2010

Submitted to: Submitted by: Ms. Priyanka Singh

Mr.

Amandeep Singh Khera Deptt. Of Mathematics Roll. No. RK6005A19 Reg.No. 11000597

Class

K6005

ACKNOWLEDGEMENT

It acknowledges all the contributors involved in the  preparation of this project. Including me, there is a hand of  my tea teacher chers, s, som some books and and inte interrnet. et. I expre xpress ss most gratitude to my subject teacher, who guided me in the right direction. The guidelines provided by her helped me a lot in completing the assignment.

The books and websites I consulted helped me to describe each each and and ever every y poin pointt ment mentio ione ned d in this this proj projec ect. t. Help Help of  orig origin inal al crea creati tivi vity ty and and illu illust stra rati tion on had had take taken n and and I have have explained each and every aspect of the project precisely.

At last it acknowledges all the members who are involved in the preparation of this project.

 Thanks AMANDEEP SINGH

ABSTRACT

After fter going oing thro throug ugh h thi this paper aper one one will ill com come across what is eigenvector and eigenvalue. What is the importance of eigen value and eigenvector in our our day to day day lif life e and hist history ory of of eigenv eigenvalu alue e and eigenv eigenvect ector or and its variou various s applic applicati ations ons in schr sc hrod odin inge gerr equa equati tion on,g ,geo eolo logy gy and and eige eigen n face faces s etc.

TABLE OF CONTENT

1.

INTRODUCTION TO EIGEN VALUE AND EIGEN VECTOR

2.

HISTORY

3. APPLICATION OF EIGEN VALUE AND EIGEN VECTOR

3.1 SCHRODINGER EQUATION 3.2 MOLECULAR ORBITAL 3.3 GEOLOGY AND GLACIOLOGY 3.4 FACTOR ANALYSIS 3.5 VIBRATION ANALYSIS 3.6 EIGEN FACES 3.7  TENSOR OF INERTIA 3.8 STRESS TENSOR 3.9 EIGEN VALUE OF A GRAPH 4.

BIBLOGRAPHY

INTRODUCTION TO EIGEN VALUE AND EIGEN VECTOR

 They are derived from the German word "eigen" which means "proper" or "characteristic." An eigenvalue of a square matrix is a scalar that is usually represented by the Greek Greek lette letterr (prono (pronounc unced ed lambda lambda). ). As you might might susp su spec ect, t, an eige eigenv nvec ecto torr is a vect vector or.. More Moreov over er,, we require that an eigenvector be a non-zero vector, in other words, an eigenvector can not be the zero vector. We will denote an eigenvector by the small letter  x . All eige eigenv nval alue ues s and and eige eigenv nvec ecto tors rs sati satisf sfy y the the equa equati tion on for a given square matrix, A matrix, A.. Consid Consider er the square square matri matrix x  A . We We sa say th that is an an of  A if there exists a non-zero vector  x  eigenvalue of  A such that . In this case,  x  is called an (corre resp spon ondi ding ng to ), and and the the pair pair ( , x   x ) is eigenvector (cor called an eigenpair for A for A..

Let's look at an example of an eigenvalue and eigenvector. If you were were asked asked if is an eigenv eigenvect ector or corres correspon pondin ding g to the eigenvalue for, you could find out by substituting x substituting x,, and A and  A into the equation

Ther Theref efor ore, e, and and  x are are an eige eigenv nval alue ue and and an eige eigenv nvec ecto tor, r,  A. respectively, for  A.

HISTORY 

Eigenvalues are often introduced in the context of linear  f linear  algebra or matr or matrix ix theory theory. Hist Histor oric ical ally ly,, howe howeve ver, r, they they aros arosee in the the study of quadratic of quadratic forms and differential differential equations. equations. Euler ha Euler had d also also stud studie ied d the the rota rotati tion onal al motio otion n of a rigid rigid body body and disc discov over ered ed the the impo import rtan ance ce of the the  princ principa ipall axes axes. As Lagrange real realiz ized ed,, the the prin princi cipa pall axes axes are are the the eige eigenv nvec ecto tors rs of the the iner inerti tiaa matrix. In the early 19th century, Cauchy saw how their work could  be used to classify the quadric surfaces, surfaces, and generalized it to arbitrary dimensions. Cauchy also coined the termracine caractéristique (characteristic (characteristic root) for what is now called eigenvalue; eigenvalue; equation . his term survives in characteristic equation. Fourier used Fourier used the work of Laplace and Lagrange to solve the heat equation by separation of variables in his famous 1822 book Théorie book Théorie analytique de la chaleur . Sturm developed Fourier's ideas further and he brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that symmetric matrices have real eigenvalue eigenvalues. s. This This was extended extended by Hermite in 1855 to what are now called Hermitia Hermitian n matrices matrices.. Arou Around nd the the same same time time,, Brioschi proved that the eigenvalues of or f orth thog ogon onal al matr matric ices es lie on the unit circle, circle, andClebsch andClebsch found the corresponding result for skew-symmetric for  skew-symmetric matrices. matrices. Finally, Weierstrass clarified an important aspect in the stabil stability ity theory theory star starte ted d by Lapl Laplac acee by real realiz izin ing g that thatdefective defective matrices can cause instability. In the meanti meantime, me, Liouville studie studied d eigenv eigenvalu aluee proble problems ms simila similarr to those of Sturm; the discipline that grew out of their work is now

Sturm-Liouville ille theory theory.. Schwarz stud called Sturm-Liouv studie ied d the the firs firstt eige eigenv nval alue ue of Laplace's of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later. At the star startt of the 20th 0th cent entury, ury, Hilbert studie studied d the eigenv eigenvalu alues es of integral of integral operators by viewing the operators as infinite matrices. He was the first to use the German word eigen to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Helmholtz. Helmholtz. "Eigen" can be translated as "own", "peculiar  to", "characteristic", or "individual" — emphasizing how important eigenvalues are to defining the unique nature of a specific trans transfor format mation ion.. For some some time, time, the standa standard rd term term in Engli English sh was "proper value", but the more distinctive term "eigenvalue" is standard today. The The firs firstt nume numeri rica call algo algori rith thm m for for comp comput utin ing g eige eigenv nval alue uess and and eigenvectors appeared in 1929, when Von Mises published the power  the power  method. method . One of the most popular methods today, the QR algorithm, algorithm, was proposed independently by John G.F. Francis and Vera Kublanovskaya in 1961. 1961.

APPLICATIONS OF EIGEN VALUES AND EIGEN VECTORS Schrödinger Equation An example example of an eigenvalu eigenvaluee equation equation where where the transfor transformati mation on T is represented in terms of a differenti ntial operator is the timeindependent Schrödinger equation in quantum mechanics: mechanics:

where H,

the Hamiltonian, Hamiltonian,

is

a

second-order  differential differential

operator an operator and d ψE, ψE, the the wavefunction, wavefunction, is one of its eigenfunctions corresponding to the eigenvalue E, interpreted as its energy. energy. Howe Howeve ver, r, in the the case case wher wheree one one is inte intere rest sted ed only only in the the bound state solutions of the Schrödinger equation, one looks for ψE within the space of square of square integrable functions. Since this space is a Hilbert space with a well-defined scalar product, product, one can introduce a basis a basis set in which ψE and H can be represented as a one-dimensional array and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form. (Fig. 8 presents the lowest eigenfunctions of the Hydrogen atom Hamiltonian.) The Dirac Dirac notation notation is ofte often n used used in this this cont contex ext. t. A vect vector or,, whic which h repr repres esen ents ts a stat statee of the the syst system em,, in the the Hilb Hilber ertt spac spacee of squa square re inte integr grab able le fun funct ctio ions ns is is repr repres esen ente ted d by Schrödinger equation is:

. In In this this not notat atio ion, n, the the

wher wheree is an eige eigens nsta tate te of H. It It is a self self adjoin adjointt operat operator  or , the infinite dimensional analog of Hermitian matrices (see Observable). Observable). As in the matrix matrix case, in the equation equation above above

is understood understood to

  be the vector obtained by application of the transformation transformation H to

.

Fig. 8. The wavefunctions asso associ ciat ated ed with with the thebound  states of an electron in a hydrogen atom can be seen as the eigenvectors of the hydrogen atom Hamiltonian as well as of the angul angular ar momen momentum tum operator . They are associated with eigenvalues interpreted as their energies (increasing downward:n=1,2,3,...) and  angular  momentum (increasing across:s, p, d,...). The illustration shows the square of the absolute value of  the the wave wavefu func ncti tion ons. s. Brigh righte terr are areas corre rresp spon ond d to higher   probabil probability ity density  density fo forr a posi positi tion onmeasurement  measurement . The center of each figure is the atom atomic ic nucl nucleu eus s, a proton.  proton.

Molecular Orbitals

In quantum quantum mechanics mechanics,, and in particular in atomic and molecular   physics,  physics, within the Hartree-Fock th Hartree-Fock theo eory ry,, the the atomic and molecular  orbitals can be defined by the eigenvectors of the Fock operator . The corresponding

eigenvalues

are

interpreted

as ionization

 potentialsvia  potentialsvia Koopmans' theorem. theorem. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explic explicitl itly y depend dependent ent on the orbita orbitals ls and their their eigenv eigenvalu alues. es. If one wants to underline this aspect one speaks of nonlinear eigenvalue  problem. Such equations are usually solved by an iteration procedure, called in this case self-consistent field method. In quantum chemistry, chemistry, one

often

represents

the

Hartree-Fock

equation

in

a

non-

orthogonal basis orthogonal basis set. set. This particular representation is a generalized eigenvalue problem called Roothaan equations. equations.

Geology And Glaciology In geology geology,, especially in the study of glacial of glacial till, till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of clasts of clasts in a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) diag diagra ram m or as a Ster Stereo eone nett on a Wulf Wulfff Net Net . The The outp output ut for for the the orientation tensor is in the three orthogonal (perpendicular) axes of  space. Eigenvectors output from programs such as Stereo32 are in the order order E1 ≥ E2 ≥ E3, with with E1 being being the primar primary y orient orientati ation on of clast clast orientation/dip, orientation/dip, E2 being the secondary and E3 being the tertiary, in terms of strength. The clast orientation is defined as the eigenvector, on a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor:

this is valued from 0° (no dip) to 90° (vertical). The relative values of E1, E2, and E3 are dictated by the nature of the sediment's fabric. If E1 = E2 = E3, the fabric fabric is said to be isotropic isotropic.. If E1 = E2 > E3the E3the fabric is planar. If E1 > E2 > E3 the fabric is linear. See 'A Practical Guide to the Study of Glacial Sediments' by Benn & Evans, 2004 .

Factor analysis In factor

analysis,

the

eigenvectors

of

a covariance

matrix or correlation or correlation matrix correspond to factors factors,, and eigenvalues to the variance explained by these factors. astatistical technique

used

in

Factor analysis

the social

is

sciences and

in marketing marketing,,  product product manag management ement, oper operation ationss rese research arch,, and other  applied sciences that deal with large quantities of data. The objective is to explain most of the covariability among a number of  observable ran random dom var variab iables les in terms of a smaller number of  unobservable latent variables called factors. The observable random vari variab able less are are mode modele led d as line linear ar comb combinat inations ions of the the fact factor ors, s, plus plus unique variance terms. Eigenvalues are used in analysis used by Qmethodology software; factors with eigenvalues greater than 1.00 are cons consid ider ered ed sign signif ific ican ant, t, expl explai aini ning ng an impo import rtan antt amou amount nt of the the variability in the data, while eigenvalues less than 1.00 are considered too weak, not explaining a significant portion of the data variability.

Vibration analysis Eige Eigenv nval alue ue prob proble lems ms occu occurr natu natura rall lly y in the the vibr vibrat atio ion n anal analys ysis is of  mechanical structures with many degrees of freedom. freedom. The eigenvalues are used to determine the natural frequencies of vibration, and the eigenvectors determine the shapes of these vibrational modes. 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. analysis.

Eigen Faces

Fig. shows eigen faces as eigen vectors

In image processing, processing, processed images of faces of faces can be seen as vectors whose components are the brightnesses the brightnesses of each pixel each pixel.. The dimension of this vector space is the number of pixels. The eigenvectors of  the covariance matrix associated to a large set of normalized pictures of faces are called eigenfaces eigenfaces;; this is an example of  principal components analysis. analysis. They are very useful for expressing any face image as a lin linear ear com combin binati ation ono of some of them. In the facial recognition bran branch ch of  biometrics,  biometrics, eige eigen nfaces aces provi rovide de a means eans of  applying dat ataa

com co mpr preess ssiion to

faces

for  identification  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

constr construct ucted. ed. These These concep concepts ts have have been been found found useful useful in automa automati ticc speech recognition systems, for speaker adaptation.

Tensor Of Inertia In mechanics mechanics,,

the

eigenvectors

of

the ine nerrtia

ten enso sor  r define define

the  the  princ princip ipal al axe axessof a ri rigi gid d bo body dy.. The tensor  tensor of  of inertia inertia is a key quantity required in order to determine the rotation of a rigid body around its center of mass.

Stress Tensor In soli solid d mech mechanics anics,, the str stress ess ten tensor  sor is is sym symmetr etric and and so can can be deco decomp mpos osed ed into into a diagonal tens tensor or with with the the eige eigenv nval alue uess on the the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation,

the

stress

tensor

has

no shear  shear co comp mpon onen ents ts;;

the the

components it does have are the principal components.

Eigenvalues Of A Graph In spectral graph theory, theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix A, or (increasingly) of the graph's Laplacian matr matrix ix,, whic which h is eith either er T−A T−A or I− I−T1 T1/2 /2AT AT −1/2 −1/2,, where T is a diagonal matrix holding the degree of each vertex, and in T −1/2, 0 is substituted for 0−1/2. The kth principal eigenvector of  a graph is defined as either the eigenvector corresponding to the kth largest eigenvalue of A, or the eigenvector corresponding to the kth smallest eigenvalue of the Laplacian. The first principal eigenvector  of the graph is also referred to merely as the principal eigenvector. eigenvector.

The The prin princi cipa pall eige eigenv nvec ecto torr is used used to meas measur uree the the centrality of its its vertices. An example is Google Google's 's PageRank  PageRank algorithm. algorithm. The principal eigenvector of a modified adjacency matrix of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the stationary distribution of the Markov chain represented by the row-n row-norm ormali alized zed adjace adjacency ncy matrix matrix;; howeve however, r, the adjace adjacency ncy matri matrix x must first be modified to ensure a stationary distribution distribution exists. The second principal eigenvector can be used to partition the graph into clusters, viaspectral viaspectral clustering. clustering. Other methods are also available for  clustering.

BIBLOGRAPHY 

1.

en.wikipedia.org/.../Eigenvalue,_eigenvector_and_eigenspace

2.

mathworld.wolfram.com › ... › Matrices › Matrix Eigenvalues

3.

www.sosmath.com/matrix/eigen0/eigen0.html

4.

www.eigenvalue.com

5.

planetmath.org/encyclopedia/Eigenvalue.html

6.

higher engineering mathematics

7.

B.V.RAMANA