Results 1 to 2 of 2

Math Help - Eigenvalues of conjugate transpose matrix

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    7

    Eigenvalues of conjugate transpose matrix

    Hello
    how do i prove that all the eigenvalues of the product A*A (where A* is the conjugate transpose matrix of A) are real non negative numbers. note A is not necessarily a square matrix ?

    Letting t be an eueigenval of A*A, with eigenvector v. Then we have A*Av = tv, and taking the inner product with v on both sides implies that v*A*Av = tv*v.

    But v*A*Av = |Av|2,

    How do I finish it off from here?

    Cheers

    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,696
    Thanks
    1467
    Quote Originally Posted by sssouljah View Post
    Hello
    how do i prove that all the eigenvalues of the product A*A (where A* is the conjugate transpose matrix of A) are real non negative numbers. note A is not necessarily a square matrix ?

    Letting t be an eueigenval of A*A, with eigenvector v. Then we have A*Av = tv, and taking the inner product with v on both sides implies that v*A*Av = tv*v.

    But v*A*Av = |Av|2,

    How do I finish it off from here?

    Cheers

    The crucial point is the basic property of the "conjugate transpose": (Au)\cdot v= u\cdot (A*v), where " \cdot is an inner product over the complex numbers: u\cdot v= \overline{v\cdot u}.
    Suppose \lambda is an eigenvalue of A*A and v is an eigenvector, corresponding to \lambda with length 1. Then
    \lambda= \lambda(v\cdot v)= (\lambda v)\cdot v= (A*Av)\cdot v = (Av)\cdot(A**v)= (Av)\cdot(Av)= v\cdot (A*Av)= v\cdot(\lambda v) = \overline{\lambda}(v\cdot v)= \overline{\lambda}.

    That is, \lambda= \overline{\lambda} and so \lambda is real.
    Last edited by mr fantastic; September 6th 2009 at 04:37 AM. Reason: Fixed a latex tag
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Matrix and Its Transpose
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 2nd 2011, 03:23 PM
  2. Conjugate Transpose Inner Product Help
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 7th 2009, 03:34 AM
  3. If matrix A is invertible then so it it transpose A^t
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 2nd 2009, 04:13 PM
  4. Replies: 2
    Last Post: February 12th 2009, 12:12 AM
  5. Finding a matrix and its transpose from a given matrix
    Posted in the Advanced Algebra Forum
    Replies: 16
    Last Post: August 10th 2007, 12:02 AM

Search Tags


/mathhelpforum @mathhelpforum