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ENGG2013 Unit 17 Diagonalization Eigenvector and eigenvalue

ENGG2013 Unit 17 Diagonalization Eigenvector and eigenvalue. Mar, 2011. EXAMPLE 1. Q6 in midterm. u(t): unemployment rate in the t-th month. e(t)= 1-u(t) The unemployment rate in the next month is given by a matrix multiplication Equilibrium: Solve.

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ENGG2013 Unit 17 Diagonalization Eigenvector and eigenvalue

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  1. ENGG2013 Unit 17Diagonalization Eigenvector and eigenvalue Mar, 2011.

  2. EXAMPLE 1 ENGG2013

  3. Q6 in midterm • u(t): unemployment rate in the t-th month. • e(t)= 1-u(t) • The unemployment rate in the next month is given by a matrix multiplication • Equilibrium: Solve  Unemployment rate at equilibrium = 0.2 ENGG2013

  4. Equilibrium Unstable Stable ENGG2013

  5. If stable, how fast does it converge to the equilibrium point? Slow convergence Fast convergence 0.2 0.2 ENGG2013

  6. Question • Suppose that the initial unemployment rate at the first month is x(1), (for example x(1)=0.25), and suppose that the unemployment evolves by matrix multiplication Find an analytic expression for x(t), for all t. ENGG2013

  7. EXAMPLE 2 ENGG2013

  8. How to count? • Count the number of binary strings of length n with no consecutive ones. ENGG2013

  9. SOLVING RECURRENCE RELATION ENGG2013

  10. Fibonacci numbers http://en.wikipedia.org/wiki/Fibonacci_number • F1 = 1 • F2 = 1 • For n > 2, Fn = Fn-1+Fn-2. • The Fibonacci numbers are • 1,1,3,5,8,13,21,34,55,89,144 ENGG2013

  11. A matrix formulation • Define a vector • Initial vector • Find the recurrence relation in matrix form ENGG2013

  12. A general question • Given initial condition and for t  2 Find v(t) for all t. ENGG2013

  13. Matrix power • Need to raise a matrix to a very high power ENGG2013

  14. A trivial special case • Diagonal matrix • The solution is easy to find • Raising a diagonal matrix to the power t is easy. ENGG2013

  15. Decoupled equations • When the equation is diagonal, we have two separate equation, each in one variable ENGG2013

  16. DIAGONALIZATION ENGG2013

  17. Problem reduction • A square matrix M is called diagonalizable if we can find an invertible matrix, say P, such that the product P–1 M P is a diagonal matrix. • A diagonalizable matrix can be raised to a high power easily. • Suppose that P–1 M P = D, D diagonal. • M = PD P–1. • Mn = (PD P–1) (PD P–1) (PD P–1) … (PD P–1) = PDn P–1. ENGG2013

  18. Example of diagonalizable matrix • Let • A is diagonalizable because we can find a matrix such that ENGG2013

  19. Now we know how fast it converges to 0.2 • The matrix can be diagonalized ENGG2013

  20. Convergence to equilibrium • The trajectory of the unemployment rate • the initial point is set to 0.1 ENGG2013

  21. EIGENVECTOR AND EIGENVALUE ENGG2013

  22. How to diagonalize? • How to determine whether a matrix M is diagonalizable? • How to find a matrix P which diagonalizes a matrix M? ENGG2013

  23. From diagonalization to eigenvector • By definition a matrix M is diagonalizable if P–1 M P = D for some invertible matrix P, and diagonal matrix D. or equivalently, ENGG2013

  24. The columns of P are special • Suppose that ENGG2013

  25. Definition • Given a square matrix A, a non-zero vector v is called an eigenvector of A, if we an find a real number (which may be zero), such that • This number  is called an eigenvalue of A, corresponding to the eigenvector v. Matrix-vector product Scalar product of a vector ENGG2013

  26. Important notes • Ifvis an eigenvector of A with eigenvalue , then any non-zero scalar multiple of v also satisfies the definition of eigenvector. k  0 ENGG2013

  27. Geometric meaning • A linear transformation L(x,y) given by: L(x,y) = (x+2y, 3x-4y) • If the input is x=1, y=2 for example, the output is x = 5, y = -5. x  x + 2y y  3x – 4y kshum

  28. Invariant direction • An Eigenvector points at a direction which is invariant under the linear transformation induced by the matrix. • The eigenvalue is interpreted as the magnification factor. • L(x,y) = (x+2y, 3x-4y) • If input is (2,1), output is magnified by a factor of 2, i.e., the eigenvalue is 2. kshum

  29. Another invariant direction • L(x,y) = (x+2y, 3x-4y) • If input is (-1/3,1), output is (5/3,-5). The length is increased by a factor of 5, and the direction is reversed. The corresponding eigenvalue is -5. kshum

  30. Eigenvalue and eigenvector of First eigenvalue = 2, with eigenvector where k is any nonzero real number. Second eigenvalue = -5, with eigenvector where k is any nonzero real number. ENGG2013

  31. Summary • Motivation: want to solve recurrence relations. • Formulation using matrix multiplication • Need to raise a matrix to an arbitrary power • Raising a matrix to some power can be easily done if the matrix is diagonalizable. • Diagonalization can be done by eigenvalue and eigenvector. ENGG2013

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