Results 1 to 2 of 2

Math Help - Proving

  1. #1
    Member
    Joined
    Apr 2008
    Posts
    191

    Proving

    (a) Show that if P is a non-singular nxn matrix then \lambda is an eigenvalue of A if and only if, \lambda is an eigenvalue of PAP^{-1}.

    (b) Show that if P is a non-singular nxn matrix such that A^3=o_{n}. Show that the only possible eigenvalue for A is \lambda = 0. Here 0_{n} is the nxn zero matrix.

    For part (a) I think I need to manipulate the expression, here's what I think:

     <br />
PAP^{-1} (x) = \lambda x<br />

    A(PxP^{-1}) = \lambda x

    A(x)PP^{-1} = A(x) I_{n} = A(x) = \lambda x


    Can anyone help me with part (b)?
    I know that lambda must satisfy Ax=\lambda x

    we have: A.A.A (x) = \lambda x

    I don't know what to do with this & I'm I'm really stuck here...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Roam View Post
    (a) Show that if P is a non-singular nxn matrix then \lambda is an eigenvalue of A if and only if, \lambda is an eigenvalue of PAP^{-1}.
    if Ax=\lambda x, put y=Px. then PAP^{-1}y=\lambda y. conversely if PAP^{-1}x=\lambda x, then put P^{-1}x=y and see that Ay=\lambda y. in each case it's important to note that x \neq 0 iff y \neq 0.


    (b) Show that if P is a non-singular nxn matrix such that A^3=o_{n}. Show that the only possible eigenvalue for A is \lambda = 0. Here 0_{n} is the nxn zero matrix.
    if Ax=\lambda x, then A^2x=A\lambda x=\lambda Ax=\lambda^2 x. repeat this to get 0=A^3x=\lambda^3x, and thus \lambda = 0, because x \neq 0.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proving?
    Posted in the Geometry Forum
    Replies: 2
    Last Post: January 24th 2011, 05:41 PM
  2. Proving the SST = SSE + SSR
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: November 9th 2010, 10:10 AM
  3. proving sin^2(x)<=|sin(x)|
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: September 22nd 2010, 01:14 PM
  4. proving
    Posted in the Number Theory Forum
    Replies: 9
    Last Post: October 25th 2009, 09:27 AM
  5. Proving an identity that's proving to be complex
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: July 21st 2009, 02:30 PM

Search Tags


/mathhelpforum @mathhelpforum