Results 1 to 5 of 5

Math Help - Symmetric matrix

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    56

    Symmetric matrix

    If A is a square matrix and x^TAy=(Ax)^Ty for all x,y\in R^n, then prove A is a symmetric matrix. ie. prove A^T=A

    I cant just go x^TAy=x^TA^Ty rite? So what should i do instead?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jun 2009
    Posts
    660
    Thanks
    133
    Use the matrix property (AB)^T=B^TA^T.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2009
    Posts
    56
    yeh i know that property but how do i do it? ive just used the property and now im at x^TAy=x^TA^Ty. But i cant just cancel them out to get A=A^T because they are matrices...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jun 2009
    Posts
    660
    Thanks
    133
    You're told that the relationship is true for all x and y, so it's certainly true for specific x and y.
    Choose x to consist of entirely of zeros apart from a 1 in the ith position and y to consist entirely of zeros apart from 1 in the jth position. Then evaluate both sides of the equation x^TA^Ty=x^TAy.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Sep 2009
    Posts
    158
    You started with:
    A is nxn
    x^TAy= (Ax)^Ty
    so
    x^TAy= x^TA^Ty
    x^TAy - x^TA^Ty = 0
    factor...
    x^T(A - A^T)y = 0
    this is true for all x,y in R^n
    consider x = 0, and y = anything in R^n we get x^T(A - A^T)y = 0
    consider y = 0, and x = anything in R^n we get x^T(A - A^T)y = 0
    consider y=x=0 we get x^T(A - A^T)y = 0
    we know the statement is true for all x,y not just x, y, being zero vectors
    the only way x^T(A - A^T)y = 0 is true for any x and y is if A- A^T = zero matrix

    hope this helped
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: April 5th 2013, 08:23 PM
  2. Symmetric relation v.s. symmetric matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 14th 2010, 11:37 PM
  3. Something about Symmetric Matrix
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: October 21st 2009, 01:35 AM
  4. symmetric matrix
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 30th 2009, 01:15 AM
  5. Symmetric Matrix
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 22nd 2008, 11:11 PM

Search Tags


/mathhelpforum @mathhelpforum