Results 1 to 3 of 3

Math Help - Characteristic Polynomial Question

  1. #1
    Newbie
    Joined
    Oct 2010
    From
    London, United Kingdom
    Posts
    7

    Characteristic Polynomial Question

    Hi,
    Really unsure about how to do this question, so any help/answers/links to a similar thread would be appreciated.

    landa is an eigenvalue of a linear map T:V to V (V over C). Suppose the eigenspace W has dimension r. Prove that the characteristic polynomial CA(x) of a nxn matrix A representing T is divisiblr by ((landa)-x)^r, and deduce that r is less than or equal to the multiplicity of landa as an eigenvalue of A.

    Thankyou
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,720
    Thanks
    1473
    If the eigenspace, W (I presume you mean "corresponding to \lambda") has dimension r, then there exist r independent vectors (a basis for W), \{v_i\}, such that Av_i= \lambda v_i. You can construct a basis for V containing those vectors: \{v_1, v_2, ..., v_r, u_1, ..., u_{n-r}\} where n is the dimension of V. Written in terms of that ordered basis, A is diagonal, with \lambda on the diagonal, for the first r rows and columns, with some other n- r by n- r matrix for the last n-r rows and columns. The Characteristic polynomial for that matrix obviously has the form (x- \lambda)^r P(x) where P(x) is an n- r degree polynomial. It is possible that x- \lambda divides P(x) but certainly it divides (x- \lambda)^rP(x) at least r times.
    Last edited by HallsofIvy; October 17th 2010 at 06:55 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2010
    From
    London, United Kingdom
    Posts
    7
    Thankyou HallsofIvy, you've been most usefull.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Characteristic Polynomial
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 21st 2010, 02:40 PM
  2. Replies: 1
    Last Post: December 15th 2009, 07:26 AM
  3. characteristic polynomial question
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: April 5th 2009, 07:55 PM
  4. characteristic polynomial
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: November 9th 2008, 11:09 AM
  5. Characteristic polynomial
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 13th 2006, 07:35 AM

Search Tags


/mathhelpforum @mathhelpforum