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[Linear Algebra] Conjugate Transpose of a Matrix and vectors in ℂ

  1. Mar 31, 2016 #1
    1. The problem statement, all variables and given/known data
    Let A be an n x n matrix, and let v, w ∈ ℂn.

    Prove that Avw = v ⋅ Aw

    2. Relevant equations
    † = conjugate transpose
    ⋅ = dot product
    * = conjugate
    T = transpose

    (AB)-1 = B-1A-1
    (AB)-1 = BTAT
    (AB)* = A*B*
    A = (AT)*
    Definitions of Unitary and Hermitian Matrices
    Complex Mod
    Vector Inner Product Space rules/axioms
    Cancellation Theorem


    3. The attempt at a solution
    (Av)w = v ⋅ (Aw)
    (Av)w = ((Aw) ⋅ v)*
    (Av) ⋅ w = ((Aw)*) ⋅v*
    (Av)w = ((A)*w*) ⋅v*
    (Av) ⋅ w = (ATw*) ⋅v*

    and thats as far as I get :)
     
    Last edited: Mar 31, 2016
  2. jcsd
  3. Mar 31, 2016 #2

    micromass

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    How is ##\mathbf{v}\cdot \mathbf{w}## defined? Can you write this as a matrix multiplication?
     
  4. Mar 31, 2016 #3

    andrewkirk

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    Write the LHS out as a double sum of matrix and vector components and rearrange until it is equal to the RHS.

    There may be a quicker way but we can't really suggest anything without knowing what theorems you are allowed to use.
     
  5. Mar 31, 2016 #4
    @micromass The LHS and RHS of the equations are defined by the vector dot product, I've made some changes in the original post in an attempt to clarify that.

    @andrewkirk I've added more to the relevant equations sections to address what you said. There are probably a few more theorems but this is what I was able to pull off the top of my head.

    Thank you!
     
  6. Mar 31, 2016 #5

    micromass

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    What is the definition of the vector dot product?
     
  7. Mar 31, 2016 #6

    andrewkirk

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    Those won't enable you to get a solution because none of them address the critical issue of what happens when you move an operator from the first part of an inner product to the second.
    Write the equation out in component form. It's only a few lines to prove it that way, and no external theorems need to be used.
     
  8. Mar 31, 2016 #7
    v ⋅ w = ∑ viwi where vi and wi are the ith entries of the vectors v and w and 1 ≤ i ≤ n where n is the last index.
     
  9. Mar 31, 2016 #8

    micromass

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    That is not the definition of the dot product since you work in ##\mathbb{C}^n##.

    If you do find the correct definition, are you able to write it as a matrix multiplication?
     
  10. Mar 31, 2016 #9
    I'll try it that way and see where I get, thanks.
     
  11. Mar 31, 2016 #10

    andrewkirk

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    Make sure you use the correct definition of the inner product in a complex vector space, as per @micromass's post above. Otherwise it won't work out.
     
  12. Mar 31, 2016 #11
    oh boy, your absolutely right. I completely forgot to add on the complex conjugate. It should be ∑vi* wi.
     
  13. Mar 31, 2016 #12
    @andrewkirk @micromass I wrote it out in component form and re-arranged it from there. Worked out perfectly.


    Thanks again!
     
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