# Doubts about the proof of $q^*q = qq^*$ property for quaternions.

by Haseo   Last Updated September 11, 2019 13:20 PM

I have a problem with the proof of this quaternions property, included below.

$$(q^∗)^∗ = [s, −(−v)] = [s, v] = q$$$$q^∗q = (q^∗)(q^∗)^∗= ∥q^∗∥^2= s^2 + x^2 + y^2 + z^2= ∥q∥^2= qq^∗$$ so we got $$q^*q = qq^*$$.

Why in the last equation we lost one of the conjugate marks?

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Conjugate operation is involutive, so $$(q^*)^* = q$$.
Thus, since $$qq^*$$ is real, $$qq^* = (qq^*)^*=q^*(q^*)^*=q^*q$$.