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Column Vectors Product

  1. Nov 24, 2009 #1
    Hello,

    Suppose that [tex]\mathbf{h}[/tex] is an [tex]N\times 1[/tex] column vector. Can we say that:

    [tex]\mathbf{h}^*\mathbf{h}=\mathbf{h}\mathbf{h}^*[/tex]

    where * means complex conjugate transpose?

    Thanks in advance
     
  2. jcsd
  3. Nov 24, 2009 #2
    Try it out for yourself. Take [tex]\mathbf{h} = [1, 0]^*[/tex].

    Furthermore, if you're not familiar with them, you may want to look up the terms inner product and outer product. They are operations on vectors that, conveniently, map to matrix multiplication between conjugate transposes.
     
  4. Nov 24, 2009 #3
    Ok then, [tex]\mathbf{h}\mathbf{h}^*[/tex] is the outer product of the two vectors, and [tex]\mathbf{h}^*\mathbf{h}[/tex] is the inner product of them. Right?
     
  5. Nov 24, 2009 #4
    Yup!

    Usually, the inner product is taken to be a real number, not a 1x1 matrix. But the two ideas are identical. The inner product is well known for it's use is defining orthogonality and angles.

    The outer product is less commonly known. It has applications when working with dual vector spaces. It's useful in QM and GR.
     
  6. Nov 24, 2009 #5
    Ok thank you Tac-Tics.

    Best regards
     
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