Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Inconsistent Inner Product Definitions

  1. Mar 18, 2006 #1

    cepheid

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Hi,

    I'm looking at the definition of the inner product of two vectors in [itex] \mathbb{C}^n [/itex]. One source is talking about how the definition of an inner product must be modified to account for vectors with complex components and says:

    He then goes on to say that we can rewrite conjugate transpose as follows: (a.k.a. the hermitian conjugate or hermitian transpose, depending which book you read, it seems. Can't we just stick to "adjoint?" :rolleyes:)

    [tex] \mathbf{\bar{x}}^{\mathrm{T}} = \mathbf{x}^{\mathrm{H}} [/tex]

    The point of this thread is that I have a second source with a contradictory definition (the second vector conjugated instead of the first):

    So what gives? Which is the correct definition? I'm inclined to believe the first one (G. Strang) if only because it is consistent with the definition given by Griffiths in his Introduction to Quantum Mechanics in Appendix A. So that's 2 sources vs. 1. Griffiths of course, uses the wacky physics notation <a|b>, which I'm still not totally used to. He also uses totally different notation for complex conjugation and the transpose, and the adjoint.
     
  2. jcsd
  3. Mar 18, 2006 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Both conventions are used. I think the "physicist" convention is antilinear in the first argument, and the "mathematician" convention is antilinear in the second argument.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Inconsistent Inner Product Definitions
  1. Inner Products? (Replies: 7)

Loading...