Results 1 to 7 of 7

Math Help - Eigenvalues and Eigenvectors, PLEASE

  1. #1
    Junior Member
    Joined
    Nov 2006
    Posts
    48

    Exclamation Eigenvalues and Eigenvectors, PLEASE

    a)Show that for any square matrix A, A^t (A transpose) and A have the same characteristic polynomial and hence the same eigenvalues.

    b)Give an example of a 2x2 matrix A for which A^t and A have different eigenspaces.

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

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by somestudent2 View Post
    a)Show that for any square matrix A, A^t (A transpose) and A have the same characteristic polynomial and hence the same eigenvalues.

    b)Give an example of a 2x2 matrix A for which A^t and A have different eigenspaces.

    Thank you.
    a) You should think about \text{det}(A - \lambda I) and \text{det}(A^T - \lambda I). Are they always the same?

    b) Shouldn't be too hard to play around with a couple of 2x2 matrices, see what happens with them ......
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2006
    Posts
    48
    would that be sufficient enough to say that since det(A) = det(A^t)

    then det(A-xI)=det(A^t-xI) since we pretty much subtract the same number, I also considered this solution except it seemed too short to me to be true?

    thanks again
    Last edited by somestudent2; January 22nd 2008 at 07:20 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by somestudent2 View Post
    would that be sufficient enough to say that since det(A) = det(A^t) Mr F says: No! det(A + B) \neq det(A) + det(B).

    then det(A-xI)=det(A^t-xI) since we pretty much subtract the same number, I also considered this solution except it seemed too short to me to be true?

    thanks again
    ..
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Nov 2006
    Posts
    48
    that is not what I meant mr F, I wrote that det(A)=det(A^t).....
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by somestudent2 View Post
    would that be sufficient enough to say that since det(A) = det(A^t)

    then det(A-xI)=det(A^t-xI) since we pretty much subtract the same number, I also considered this solution except it seemed too short to me to be true?

    thanks again
    "then det(A-xI)=det(A^t-xI) since we pretty much subtract the same number .." is waaaay too vague ... you need to show, using a mathematical argument, how det(A) = det(A^t) leads to det(A-xI)=det(A^t-xI) ......
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by somestudent2 View Post
    would that be sufficient enough to say that since det(A) = det(A^t)

    then det(A-xI)=det(A^t-xI) since we pretty much subtract the same number, I also considered this solution except it seemed too short to me to be true?

    thanks again
    Note 1: (A - \lambda I)^T = A^T - \lambda I^T = A^T - \lambda I.

    Note 2: \text{det} (A - \lambda I) = \text{det} (A - \lambda I)^T using \text{det} B = \text{det} B^T.

    But (A - \lambda I)^T = A^T - \lambda I.

    Therefore \text{det} (A - \lambda I)^T = \text{det} (A^T - \lambda I).

    Therefore ......
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. eigenvalues and eigenvectors
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 21st 2011, 12:23 PM
  2. Eigenvalues/Eigenvectors
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 20th 2010, 05:31 AM
  3. Eigenvalues and Eigenvectors
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: November 16th 2009, 06:00 AM
  4. eigenvalues and eigenvectors
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: October 1st 2009, 02:42 AM
  5. Eigenvalues/Eigenvectors
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 7th 2009, 10:15 PM

Search Tags


/mathhelpforum @mathhelpforum