1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Minimal polynomial

  1. Oct 12, 2013 #1
    1. The problem statement, all variables and given/known data
    Given the matrix
    2 0 0 0 0 0 0
    1 2 0 0 0 0 0
    0 1 2 0 0 0 0
    0 0 1 2 0 0 0
    0 0 0 0 2 0 0
    0 0 0 0 1 2 0
    0 0 0 0 0 0 2

    What is the minimal polynomial?

    2. Relevant equations

    -

    3. The attempt at a solution

    This is the Jordan form, so I guess the solution is just m(t) = (t-2)7 but I don't know if it's right. Can anyone help me?
     
  2. jcsd
  3. Oct 12, 2013 #2

    I like Serena

    User Avatar
    Homework Helper

    Hi cristina89! :smile:

    The minimal polynomial P of a square matrix A is the unique monic polynomial of least degree, m, such that P(A) = 0.

    The degree of the minimal polynomial is determined by the size of the largest Jordan block, which is 4 in your case.
    So the minimal polynomial is m(t) = (t-2)4.

    Indeed ##(A-2I)^4=0##.
     
  4. Oct 12, 2013 #3
    Thank you so much!! :)
     
  5. Oct 12, 2013 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    The method suggested above is by far the simplest way to deal with this specific problem, but in a more general case you can use the algorithms employed by computer algebra systems, such as Maple: regard A, A^2, A^3,... as n^2-dimensional vectors, then find the smallest k such the vectors I, A, A^2,..,A^k are linearly dependent---essentially, using standard linear algebra methods. This will also deliver the coefficients and hence the minimal polynomial.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Minimal polynomial
  1. Minimal Polynomial (Replies: 7)

  2. The minimal polynomial (Replies: 2)

Loading...