THE EIGENVALUE PROBLEM

THE EIGENVALUE PROBLEMBYYAN RU LINSCOTT HENDERSONNIRUPAMA GOPALASWAMIGROUP 411.1 EIGENVALUES & EIGENVECTORS

DefinitionD.C. Lay, "Eigenvectors and Eigenvalues," in Linear Algebra and Its Applications, 3rd ed. Boston, MA: Pearson, 2006, ch. 5, pp. 301-372FormulationEssentially, for some value of given a transformation matrix A, there may exist a vector x such that the equation is satisfied. If and this particular vector x do exist, then we call the eigenvalue and the x the corresponding eigenvector.ExampleImportanceEigenvalues and eigenvectors find numerous applications in these areas:Differential EquationsDynamical systemsEngineering designChemistry and physicsSchrdinger equation (quantum mechanics)Vibration analysisBRAINBITE http://www.maths.usyd.edu.au/u/UG/JM/MATH1014/Quizzes/quiz10.html

Answer: CA. =2B. =0C. =-3D. =111.2 EIGENVALUES SOLUTION PROCEDURE AND APPLICATIONS

Ax = lx (A-lI)x = 0x=0 is a trivial solutionNon-trivial solutions exist if and only if:

11.2 Eigenvalues Solution Procedure and ApplicationsResulting algebraic equation is called the characteristic equation.Characteristic polynomial- nth-order polynomial in lRoots are the eigenvalues {l1, l2, , ln}Solution space is called eigenspace corresponding to {l1, l2, , ln} The solutions obtained are called eigenvectors

Eigenvalue ExampleCharacteristic matrix

Characteristic equation

Eigenvalues: l1 = -5, l2 = 2

11.2 Subsection(1) -Quick TipsAn n x n matrix A means that are n values to x, and there will be n eigenvectors and eigenvalues even if some are duplicatedThe eigenvalues of a triangular matrix are the entries on its main diagonal Consider that since is scalar, A must act on eigenvectors only to stretch x and not to change its direction (see figure)

Unknown. (2011, Oct 27).Eigenvalues and eigenvectors[Online]. Available:http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectorsExampleClick here to view a demo on eigenvalues and eigenvectorshttp://web.mit.edu/18.06/www/Demos/eigen-applet-all/eigen_sound_all.html

11.2 Subsection(2)-Determining EigenvectorsFirst determine eigenvalues: {l1, l2, , ln}Then determine eigenvector corresponding to each eigenvalue:

Eigenvectors determined up to scalar multipleDistinct eigenvaluesProduce linearly independent eigenvectorsRepeated eigenvaluesProduce linearly dependent eigenvectorsIf n roots are equal then the eigenvalues are said to of multiplicity n.

Eigenvector ExampleEigenvalues

Determine eigenvectors: Ax = lx

Eigenvector for l1 = -5

Eigenvector for l1 = 2

BRAINBITE

http://www.maths.usyd.edu.au/u/UG/JM/MATH1014/Quizzes/quiz12.htmlAnswer : c

11.2.2 APPLICATIONS TO ELEMENTARY SINGULARITIES IN THE PHASE PLANEConsider a linear system of ODEs given by

If the eigenvalues is real

Criteria Type < 0

Stable node > 0

Unstable node

> 0 and < 0Saddle Criteria Typea < 0

Stable focusa = 0

Centre

a > 0Unstable focus 11.2. subsection(3)Special matrices in exercises(1) Markov Matrix Let

A=

The sum of elements of row or column sum to unity.One of the eigenvalue of Markov matrix is 1.The rows of [A-I]sum to zero [A-I] is singular and columns of A-I are linearly dependent.

M.D. Greenberg, "The Eigenvalue Problem," inAdvanced Engineering Mathematics, 2nd ed. Upper Saddle River, New Jersey: Prentice Hall, 1998, ch. 11.3(2)Tridiagnol matrix A Tridiagnol matrix is one in which all element are zero except the principal diagonals and its two adjacent diagonals .

Eigenvalues are given by

(3) Generalized eigenvalue problemIf B1 then Ax=Bx is called generalized eigenvalue problem.Characteristic equation got by det(A - B)x=0 Eigenvectors given by (A - B)x=0 (4) Cayley hamilton theoremTheorem- The characteristic equation of any square matrix A is n+ 1 n-1 +. n =0 then An+ 1 An-1++ n -1A+ n I=0.

i.e A satisfies characteristic equation.

BRAINBITE

http://www.maths.usyd.edu.au/u/UG/JM/MATH1014/Quizzes/quiz12.htmlAnswer : a

11.3 SYMMETRIC MATRICES[1] http://www.aiaccess.net/English/Glossaries/GlosMod/e_gm_symmetric_matrix.htm[2]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.3, pp. 354-362.

[1]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.3, pp. 354-362.[1]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.3, pp. 354-362.Let A be a real symmetric matrix. Then eigenvectors associated with distinct eigenvalues are orthogonal.Let A be a real symmetric matrix. Then there is a real, orthogonal matrix that diagonalizes A[1]. Let A be a real, n x n symmetric matrix. Then its eigenvector provide an orthogonal basis for n-space. Therefore, if an eigenvalue is repeated by k times. Then the eigenspace is of dimension k, and we can find another set of orthogonal vector by linear combination[2].

[1] Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.3, pp. 354-362.[2] M.D. Greenberg, "The Eigenvalue Problem," inAdvanced Engineering Mathematics, 2nd ed. Upper Saddle River, New Jersey: Prentice Hall, 1998, ch. 11.3, pp. 554-569.

Symmetric Matrix Examples

We can see that a real, symmetric matrix provides a set of real eigenvalues. And the corresponding eigenvectors are

These form an orthogonal set of vectors[1].

[1]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.3, pp. 354-362.The orthonormal form is divided by its length and they can be used as columns of an orthogonal matrix.

We can find Q-1 = QT, and A can be diagonalized by Q[1].

[1]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.3, pp. 354-362.Useful properties[1] [1]http://www.math.panam.edu/BRAINBITE

[1]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.3, pp. 354-362. answer : d11.4 DIAGONALIZATIONBackgroundProperties and RestrictionsExampleExampleExampleDiagonalizationExample11.5 Applications to first order systems with constant coefficients11.5 Applications to first order systems with constant coefficientsConsider an initial value problem

In matrix form

The solution to the differential equation is given by

Where

A= coefficients of variablesQ= modal matrix =[e1,e2.en]D= Diagonal matrix where jth diagonal elements are jth eigenvalue of A.The solution can also be expressed of the form

Where

M.D. Greenberg, "The Eigenvalue Problem," inAdvanced Engineering Mathematics, 2nd ed. Upper Saddle River, New Jersey: Prentice Hall, 1998, ch. 11.3ExampleConsider the equations

Solution :

Replacing the values of A,D,Q and Q-1 in the following equation

we get

11.6 QUADRATIC FORMS[1]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.4, pp. 363-367.[1]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.4, pp. 363-367.Quadratic forms example[1]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.4, pp. 363-367.Classication of The Quadratic Form[1] [1]http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=0CGQQFjAE&url=http%3A%2F%2Fwww.econ.iastate.edu%2Fclasses%2Fecon501%2FHallam%2Fdocuments%2FQuad_Forms_000.pdf&ei=wp26TvrgMsiJsAKNzqXOCA&usg=AFQjCNGQ_OibQn6rhf0wrBTNSVMVOltoaQ&sig2=Y061Hf2_fqbXqjUYyADczQClassication example[1][1]http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=0CGQQFjAE&url=http%3A%2F%2Fwww.econ.iastate.edu%2Fclasses%2Fecon501%2FHallam%2Fdocuments%2FQuad_Forms_000.pdf&ei=wp26TvrgMsiJsAKNzqXOCA&usg=AFQjCNGQ_OibQn6rhf0wrBTNSVMVOltoaQ&sig2=Y061Hf2_fqbXqjUYyADczQGraphical Analysis[1][1]http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=0CGQQFjAE&url=http%3A%2F%2Fwww.econ.iastate.edu%2Fclasses%2Fecon501%2FHallam%2Fdocuments%2FQuad_Forms_000.pdf&ei=wp26TvrgMsiJsAKNzqXOCA&usg=AFQjCNGQ_OibQn6rhf0wrBTNSVMVOltoaQ&sig2=Y061Hf2_fqbXqjUYyADczQGraphical Analysis[1]

[1]http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=0CGQQFjAE&url=http%3A%2F%2Fwww.econ.iastate.edu%2Fclasses%2Fecon501%2FHallam%2Fdocuments%2FQuad_Forms_000.pdf&ei=wp26TvrgMsiJsAKNzqXOCA&usg=AFQjCNGQ_OibQn6rhf0wrBTNSVMVOltoaQ&sig2=Y061Hf2_fqbXqjUYyADczQ[1]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.4, pp. 363-367.[1]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.4, pp. 363-367.Principal Axis Example[1]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.4, pp. 363-367.

[1]Peter V. ONeil, Eigenvalues,Diagnolization and Special Matrices in Advanced Engineering Mathematics 5th edition. Birmingham, AL: B. Stenquist, 2003, ch. 8.4, pp. 363-367.