Results 1 to 7 of 7

Math Help - Help please. linear algebra university questions.

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    3

    Help please. linear algebra university questions.

    Linear algebra question

    Let A be a n x n symmetric matrix, and let L be an eigenvalue of A with corresponding eigenvector X.

    Show that L is also an eigenvalue of the matrix P(transposed)AP where P is an orthogonal matrix. State the corresponding eigenvector of P(transposed)AP.

    How is the result modified if P(transposed)AP is a diagonal matrix D?

    I have no idea even where to start. Thanks guys.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    By using the definition of eigenvalue and the property of orthogonal matrix, It is easy to prove that L is the eigenvalue of the matrix P(transposed)AP when P is an orthogonal matrix(note that P(transposed)P=E).
    the corresponding eigenvector of P(transposed)AP is P(transposed)X.
    If P(transposed)AP is a diagonal matrix D,then D gives all the eigenvalue of A, and the corresponding eigenvector of each eigenvalue coincide with the corresponding column vectors of P.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by rebecca2812 View Post
    Linear algebra question

    Let A be a n x n symmetric matrix, and let L be an eigenvalue of A with corresponding eigenvector X.

    Show that L is also an eigenvalue of the matrix P(transposed)AP where P is an orthogonal matrix. State the corresponding eigenvector of P(transposed)AP.

    How is the result modified if P(transposed)AP is a diagonal matrix D?

    I have no idea even where to start. Thanks guys.

    1) As P is a orthogonal matrix it is invertible and in fact P^t=P^{-1} .

    2) Being P invertible there exists a vector u s.t. Pu=x\Longleftrightarrow u=P^{-1}x .

    Well, now using that all the maps here are linear, show that u is an eigenvector of P^tAP corresponding to the eigenvalue \lambda
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Nov 2009
    Posts
    3
    I still don't understand the proof for the first part. I'm never going to pass this module, I can't even understand it when people spell it out to me. I hate matrices.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by rebecca2812 View Post
    I still don't understand the proof for the first part. I'm never going to pass this module, I can't even understand it when people spell it out to me. I hate matrices.

    Well, that "optimistic' attitude won't make a lot for you, either. You've plenty of books, internet resources, etc. to try to understand.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,444
    Thanks
    1863
    Quote Originally Posted by rebecca2812 View Post
    Linear algebra question

    Let A be a n x n symmetric matrix, and let L be an eigenvalue of A with corresponding eigenvector X.

    Show that L is also an eigenvalue of the matrix P(transposed)AP where P is an orthogonal matrix. State the corresponding eigenvector of P(transposed)AP.

    How is the result modified if P(transposed)AP is a diagonal matrix D?

    I have no idea even where to start. Thanks guys.
    You know that AX= LX. Let Y= P^TX= P^{-1}X Then X= PY and so AX= A(PY)= APY= LX. Now Apply P^T to both sides of APY= LX.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Nov 2009
    Posts
    3

    Smile

    thanks so much, that was what I needed, have done my whole cw now. cheers everyone.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. 2 Linear algebra questions
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: January 7th 2012, 03:06 AM
  2. university 1st yr algebra--congruence
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: November 3rd 2008, 08:24 PM
  3. linear algebra questions
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: June 28th 2008, 10:25 PM
  4. Help with University Algebra Problems
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: October 2nd 2007, 06:42 PM
  5. University Algebra Problem
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: September 11th 2006, 08:26 PM

Search Tags


/mathhelpforum @mathhelpforum