1 / 19

ENGG2013 Unit 18 The characteristic polynomial

ENGG2013 Unit 18 The characteristic polynomial. Mar, 2011. Linear Discrete-time dynamical system. Three objects are required to specifie a linear discrete-time dynamical system. State vector u (t): a vector of length n, which summarizes the status of the system at time t.

Download Presentation

ENGG2013 Unit 18 The characteristic polynomial

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. ENGG2013 Unit 18The characteristic polynomial Mar, 2011.

  2. Linear Discrete-time dynamical system Three objects are required to specifie a linear discrete-time dynamical system. • State vector u(t): a vector of length n, which summarizes the status of the system at time t. • Transitional matrix A: how to obtain the state vector u(t+1) at time t+1 from the state vector u(t) at time t. u(t+1) = Au(t). • Initial state u(0): the starting point of the system. ENGG2013

  3. Example • The unemployment rate problem in midterm • u(t) is the unemployment rate in the t-th month and e(t) is 1-u(t). ENGG2013

  4. Last time • 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 the eigenvalue of A corresponding to the eigenvector v. Matrix-vector product Scalar product of a vector ENGG2013

  5. Another geometric picture • Take for example. • Define a recursion by u(t+1) = A u(t) • The initial vector is u(0) = [0 0.5]T. u(9) u(8) u(7) u(6) u(0) ENGG2013

  6. Dependency on initial condition • Define u(t+1) = A u(t) • The initial vector is u(0) = [1 -3]T. u(1) u(5) u(0) u(6) u(7) u(8) u(9) ENGG2013

  7. Eigenvector • An eigenvector of is a nonzero vector v such that if we start from u(t) = v,we will stay on the linewith direction v. ENGG2013

  8. A recipe for calculating eigenvalue  is an eigenvalue of A • A x =  x for some nonzero vector x • A x =  Ix for some nonzero vectorx • (A –  I ) x = 0 has a nonzero solution • A –  I is not invertible • det ( A –  I ) = 0 ENGG2013

  9. Characteristic polynomial • Given a square matrix A, if we expand the determinant the result is a polynomial in variable , and is called characteristic polynomial of A. • The roots of the characteristic polynomial are precisely the eigenvalues of A. ENGG2013

  10. First eigenvalue of • Eigenvalue = 1.5, the corresponding eigenvector is where k is anynonzero constant. • The initial point u(0)is somewhere on theline y = x. u(0) ENGG2013

  11. Another eigenvalue of • Eigenvalue = 0.5, the corresponding eigenvector is where k is anynonzero constant. u(0)=(-10,10) u(1) u(2) ENGG2013

  12. The direction [-1 1]T is not stable • In this example, if we start from a point very close to the line y= –x, for example,if the initial point isu(0)=(-9.9, 10), it will diverge. u(0) u(1) u(2) ENGG2013

  13. The direction [-1 1]T is not stable • If we start from another point very close to the line y= –x, u(0) = say (-10, 9.9), it will also diverge. u(0) u(1) u(2) ENGG2013

  14. Fibonacci sequence • F1 = 1, F2 = 1, and for n > 2, Fn = Fn–1+Fn–2. • The Fibonacci numbers are 1,1,3,5,8,13,21,34,55,89,144,… • Define a vector • The recurrence relation in matrix form ENGG2013

  15. How to find F1000 without going through the recursion? • F1000 also counts the number of binary strings of length 1000 with no consecutive ones. • We need a closed-form formula for the Fibonacci numbers. ENGG2013

  16. Closed-form formula • An expressions involving finitely many + –  , and some well-known functions. • http://en.wikipedia.org/wiki/Closed-form_expression • Integral, infinite series etc. (anything which involves the concept of limit in calculus) are not allowed. • For example, the roots of x2+x+1= 0 can be written in closed-form expression, namely ENGG2013

  17. Closed-form formula (cont’d) • By the theory of Abel and Galois, a polynomial in degree 5 or higher in general has no closed-form formula. • The function has no closed-form formula • Geometric series 1+x+x2+x3+… has closed-form formula 1/(1– x) if |x|<1. ENGG2013

  18. Niels Henrik Abel http://en.wikipedia.org/wiki/Niels_Henrik_Abel • 5 August 1802 – 6 April 1829 • Norwegian mathematician • Gave the first rigorous proofthat quintic equation in generalcannot be solved using radical. ENGG2013

  19. Évariste Galois • October 25, 1811 – May 31, 1832 • French mathematician • Tell us precisely under what condition aa polynomial is solvable using radical. • Galois theory of equations. http://en.wikipedia.org/wiki/Galois ENGG2013

More Related