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Adjoint magic? Absurdity from nothing?

  1. Apr 25, 2010 #1
    Let me put this here, as it's so simple that everyone can take a look at it. I must be blind not to see my misconclusion, but look for yourself

    Let [tex]a^TA^\dag b := (a^TAb)^*[/tex] be the definition of the adjoint [tex]A^\dag[/tex].

    I thus conclude that

    [tex] \underbrace{a^T A^\dag }_{\tilde a^T} \underbrace{B^\dag b}_{\tilde b} = \tilde a^T B^\dag b \stackrel{\text{ex vi termini}}{=}(\tilde a^T B b)^* = (a^T A^\dag B b)^*[/tex]
    [tex]\ldots \qquad= a^T A^\dag \tilde b \stackrel{\text{ex vi termini}}{=}(a^T A \tilde b)^* = (a^T A B^\dag b)^*[/tex]

    Which is generally true and thus [tex]A^\dag B = A B^\dag[/tex] which is nonsense!?

    Edit: Darn, I got the definition of the adjoint wrong, so my conclusion is correct.
    Last edited: Apr 25, 2010
  2. jcsd
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