# Adjoint magic? Absurdity from nothing?

1. Apr 25, 2010

### ManDay

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 $$a^TA^\dag b := (a^TAb)^*$$ be the definition of the adjoint $$A^\dag$$.

I thus conclude that

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

Which is generally true and thus $$A^\dag B = A B^\dag$$ which is nonsense!?

Edit: Darn, I got the definition of the adjoint wrong, so my conclusion is correct.

Last edited: Apr 25, 2010