Results 1 to 2 of 2

Math Help - Hermitian matrices

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    21

    Hermitian matrices

    Can you, please help me with this:
    (a) Show that if C is in Mn(C) is Hermitian and x*Cx = 0 for all x is in Cn, then C = 0.
    (b) Show that for any A is in Mn(C) there are ( unique ) Hermitian matrices B and C for which A = B + iC.
    (c) Show that if x*Ax is real for all x is in Cn, then A is Hermitian.
    Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by mivanova View Post
    (a) Show that if C is in Mn(C) is Hermitian and x*Cx = 0 for all x is in Cn, then C = 0.
    (b) Show that for any A is in Mn(C) there are ( unique ) Hermitian matrices B and C for which A = B + iC.
    (c) Show that if x*Ax is real for all x is in Cn, then A is Hermitian.
    For (a) I would use a polarisation identity, y^*Cx = \tfrac14\bigl((x+y)^*C(x+y) + i(x+iy)^*C(x+iy) - (x-y)^*C(x-y) i(x-iy)^*C(x-iy)\bigr). If x^*Cx=0 for all x, then each term the right side of the identity will be 0, and so y^*Cx=0 for all x and y. It easily follows that C=0. Note that this result is true for all nn complex matrices, they don't need to be hermitian.

    (b) is kind of obvious. Just take B=\tfrac12(A+A^*),\ C=\tfrac1{2i}(A-A^*). (Of course, you have to check that B and C are hermitian.)

    For (c), choose some particular values for x. If x has a 1 for the j'th coordinate and zeros everywhere else then x*Ax is just the (j,j)-element of A. So if that is real then A has real elements on the diagonal. Now take x to have two nonzero elements, in the j'th and k'th coordinates. If both these coordinates are 1, then the condition x*Ax is real tells you that the imaginary part of the sum of the (j,k)- and (k,j)-elements is zero. If the j'th coordinate of x is 1 and the k'th coordinate is i, it tells you that the real parts of the (j,k)- and (k,j)-elements of A are the same. Put those results together and they tell you that A is hermitian.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. problem involving hermitian matrices
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 16th 2011, 09:36 AM
  2. Proofs on Hermitian matrices.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 16th 2011, 05:52 PM
  3. two Hermitian problems
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 11th 2011, 05:06 AM
  4. Hermitian operators
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: December 17th 2010, 02:54 AM
  5. [SOLVED] Hermitian, Unitary, and Involutory Matrices
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: November 9th 2009, 10:02 AM

Search Tags


/mathhelpforum @mathhelpforum