Results 1 to 3 of 3

Math Help - Eigenvalues

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    24

    Wink Eigenvalues

    Hi, can you help me with this:
    Define T is in F^3 by TX=AX, where
    A= (1 0 0
    1 1 1
    1 -1 1);
    i) If F=R (reals), determine all eigenvalues and eigenvectors of T;
    i) If F=C (complexes), determine all eigenvalues and eigenvectors of T;
    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,393
    Thanks
    1327
    Quote Originally Posted by bamby View Post
    Hi, can you help me with this:
    Define T is in F^3 by TX=AX, where
    A= (1 0 0
    1 1 1
    1 -1 1);
    i) If F=R (reals), determine all eigenvalues and eigenvectors of T;
    i) If F=C (complexes), determine all eigenvalues and eigenvectors of T;
    Thank you!
    An eigenvalue of a matrix, A, is a value \lambda such that Av= \lambda v for some non-zero vector v. That is the same as Av- \lambda v= (A- \lambda I)v= 0. Since v= 0 is an obvious solution to that problem, we are saying that this must not have a unique solution and so A- \lambda I must not have an inverse. A condition that it not have an inverse is that its determinant is 0. So we have the "characteristic equation" which, for this matrix, is
    \left|\begin{array}{ccc}1-\lambda & 0 & 0 \\ 1 & 1-\lambda & 1 \\ 1 & -1 & 1-\lambda \end{array}\right|= 0
    Expanding on the first row, that is
    (1- \lambda)\right|\begin{array}{cc}1-\lambda & 1 \\ -1 & 1-\lambda \end{array}\right|= 0
    (1-\lambda)((1-\lambda)^2+ 1= (1-\lambda)^3+ (1-\lambda)= 0
    i) What are the real roots of that? Can you find the eigenvector?

    ii) What are the complex roots of that? can you find the eigenvectors?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2008
    Posts
    24

    Eigenvalues


    i) What are the real roots of that? Can you find the eigenvector?
    1 is the real root, (a,b,c)=(0,0,0) is eigenvector
    ii) What are the complex roots of that? can you find the eigenvectors?
    1+2i, 1-2i are the complex roots, x=(0, L, 2iL), x=(0, L, -2iL) are eigenvectors

    Is that right? I am not sure about (a,b,c)=(0,0,0)

    Thank you!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 21st 2010, 12:57 PM
  2. Eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 21st 2009, 12:38 PM
  3. eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 13th 2008, 07:43 AM
  4. eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 12th 2008, 10:12 PM
  5. Eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: October 12th 2008, 07:25 PM

Search Tags


/mathhelpforum @mathhelpforum