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

Dot/bilinear product in C^n / Orthogonality

  1. Jul 29, 2011 #1
    Say I have 2 complex (normalized) column vectors x and y in C^N:

    The standard dot product <x,y> = x*y (where * denotes conjugate transpose) gives me a "measure of orthogonality" of the two vectors.

    Now the bilinear product (c,y) = x'y (' denotes transpose) seems to give another "measure of orthogonality" for a somehow 'weaker' notion of orthogonality..

    Can somebody point me in any direction to better grasp this concept, i'm having a hard time understanding this second "measure" ?
  2. jcsd
  3. Jul 29, 2011 #2
    That is exactly one of the reasons why you need to take conjugates on complex vectors to have a valid way of defining an inner product on a vector space over C;
    your second product (x,y) = x'y is not an inner product over C^n, because it is not positive definite ( take the norm of ( i , 0 ) for example ). As you can see, if it is not positive definite, it fails to be an inner product.
    If you didn't know, or have forgotten, positive-definite means that your form satisfies <x,x> > 0 for all x unless x = 0, in which case <x,x> = 0 must be true.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook