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?

Answers 1

Conjugate operation is involutive, so $(q^*)^* = q$.

Thus, since $qq^*$ is real, $qq^* = (qq^*)^*=q^*(q^*)^*=q^*q$.

Vasily Mitch
Vasily Mitch
September 11, 2019 13:14 PM

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