1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Hermitian Conjugate

  1. Dec 29, 2007 #1
    Simple question, and pretty sure I already know the answer - I just wanted confirmation,

    Considering the Hermitian Conjugate of a matrix, I understand that

    [tex]A^{+} = A[/tex] where [tex]A^{+} = (A^{T})^{*}[/tex]

    Explicitly,

    [tex](A_{nm})^{*} = A_{mn}[/tex]

    Would this mean that for a matrix of A, where A is

    a b
    c d

    that

    a b
    c d

    =

    a* c*
    b* d*

    =

    A11 A12
    A21 A22

    =

    A11* A21*
    A12* A22*

    Thanks for the clarification!
     
  2. jcsd
  3. Dec 29, 2007 #2
    And can I also ask why this seems to be a general property of the Hermitian Conjugate?

    [tex](AB)^{+} = B^{+} A^{+}[/tex]

    rather than

    [tex](AB)^{+} = A^{+} B^{+}[/tex]
     
  4. Dec 29, 2007 #3

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

    for your first post, you have done correct.

    a b
    c d

    becomes

    a* c*
    b* d*

    when you do hermitian conjugate of it.

    And
    [tex](AB)^{\dagger} = B^{\dagger} A^{\dagger}[/tex]

    Follows from
    [tex](AB)^{T} = B^{T} A^{T}[/tex]

    Very easy to prove
     
  5. Dec 29, 2007 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    As for
    [tex](AB)^{\dagger} = B^{\dagger} A^{\dagger}[/tex]
    and
    [tex](AB)^{T} = B^{T} A^{T}[/tex]

    remember that multiplication of matrices is NOT commutative.
    With [itex](AB)^{T} = B^{T} A^{T}[/itex] we have [itex](AB)^T(AB)= (A^T)(B^T B)(A)= A^T A= I[/itex]. If we tried, instead, [itex](A^TB^T)(AB)[/itex] we would have [itex](A^T)(B^T A)(B)[/itex] and we can't do anything with that.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Hermitian Conjugate
  1. Hermitian problem (Replies: 0)

  2. Hermitian Operators (Replies: 1)

  3. Hermitian Identity (Replies: 3)

  4. Hermitian matrices (Replies: 2)

Loading...