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Unitary matrix

  1. Feb 1, 2009 #1
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    1. show that the determinant of a unitary matrix is a complex number of unit modulus







    2. i know the equation for a determinant, but i guess to i am not sure what a complex number of unit modulus is either. I'm looking for guidance
     
    Last edited: Feb 2, 2009
  2. jcsd
  3. Feb 1, 2009 #2

    gabbagabbahey

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    The modulus of a complex number [itex]z=x+iy[/itex], where [itex]x[/itex] and [itex]y[/itex] are real numbers representing the projections of [itex]z[/itex] onto the real and imaginary axes respectfully, is simply given by [itex]|z|=\sqrt{x^2+y^2}[/itex].

    So a complex number with unit modulus is simply a complex number [itex]z[/itex] such that [itex]|z|=\sqrt{x^2+y^2}=1[/itex].

    To find the determinant of a unitary matrix, start with the definition of unitary matrices (in the form of an equation) and take the determinant of both sides of the equation.
     
  4. Feb 1, 2009 #3
    well my problem gives the matrix of [[a,b][c,d]] and gives the det([[a,b][c,d]])=ad-bc

    then states the question i gave above.

    i read that the |det(unitary matrix)|=1, but isn't that what i am trying to solve for.

    and i am not sure if i have seen the definition of unitary matrices in the form of an equation.

    right now this is for a high level undergrad quantum course which i have to take self paced and this is my first hurdle.
     
  5. Feb 1, 2009 #4
    this is my attempt i just thought about. call the matrix R

    abs(det(R x R*))=1 since R x R* is I and det(I) = 1

    and then abs(det(R) x det(R*))=1

    and i get to a^2d^2+b^2c^2=1

    but i don't know if that does anything for me
     
  6. Feb 2, 2009 #5

    gabbagabbahey

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    You seem to be starting with the result you are trying to prove....looks like circular logic to me...

    This is the definition of a unitary matrix, and this is what you should start with.


    So start with [tex]RR^{*}=I[/tex] and take the determinant of both sides....their is a rule for taking the determinant of a product of matrices, and a rule for taking the determinant of the conjugate transpose of a matrix...use those rules!:smile:
     
  7. Feb 2, 2009 #6
    thanks for the help. i think i have it now.

    i think i had it a while ago but didn't reason it to myself right.

    i tried to prove a little more than i had to.
     
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