Results 1 to 2 of 2

Math Help - Simple eigenvalue / core-nilpotent decomposition

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    4

    Simple eigenvalue / core-nilpotent decomposition

    I am reading a book on linear algebra, and I found the following statement:

    " Assume z is a simple eigenvalue of the matrix A. Because z is a simple eigenvalue, the core-nilpotent decomposition insures that A-zI is similar to a matrix of the form:
    C 0
    ( ). "
    0 01x1

    I have trouble seeing why this is the case. This statement is used in the book to prove that the intersection between Range of (A-zI) and Nullspaceof (A-zI) is empty (so index(A-zI) = 1 and y*x <> 0 (where y is a left-hand eigenvector and x is an eigenvector of A). So I can't use these facts as assumptions to construct my argument!

    Many thanks for your help!!!

    Damien
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Sep 2009
    Posts
    4
    Hi everyone,

    On this question, someone was kind enough to suggest the following outline of a proof.

    1. Since z is a simple eigenvalue A, dim Nullspace(A-zI) = 1, i.e. there is a single linearly independent eigenvector associated with A-zI.

    Let's call this vector x and let's complete it with n-1 vector x1, x2, ..., xn-1, xn = x so that the n vectors form a basis for my vector space (Rn or Cn). (Notice that the eigenvector is placed in the last column of P).

    2. Place the n vectors as column in the matrix P: P = (x1|x2|...|xn). Now rank(P) = n so P is non-singular.

    3. Show the form of P^(-1)AP. I use block matrix multiplication, so:

    P^(-1)AP = transpose((y1|y2|...|yn)) A (x1|x2|...|xn) =

    [y1Ax1 y1Ax2 ... y1Axn
    y2Ax1 y2Ax2 ... y2Axn
    ...
    ynAx1 ynAx2 ... ynAxn]

    I can see straight away that the last column is 0nx1 since Axn = 0, but I fail to see why the last row should be 0nx1 too.

    I would really appreciate any help here.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Simple eigenvalue problem (x'=[2x2]x)
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: March 25th 2011, 11:03 AM
  2. Core 2 mock exam....
    Posted in the Math Topics Forum
    Replies: 0
    Last Post: February 15th 2010, 03:20 PM
  3. Proof: every Hermitian matrix has eigenvalue decomposition
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 16th 2009, 11:53 AM
  4. Eigenvalue (Shouldn't it be simple?)
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 9th 2007, 09:34 AM
  5. Decomposition into Finite Simple Groups
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: November 28th 2005, 11:17 PM

Search Tags


/mathhelpforum @mathhelpforum