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Anyone know an easy way to show that multiplication by a unit quaternion is a unitary operator?
by edflsafoiewq 8y ago
Anyone know an easy way to show that multiplication by a unit quaternion is a unitary operator?
- joppy 8y agoUnitary in what sense? Clearly multiplication by a unit quaternion preserves the quaternion norm.
- edflsafoiewq 8y agoIn the sense of preserving the dot product, x.y = (ux).(uy).
- joppy 8y agoSince the quaternion norm is induced via the quaternion dot product, any transformation that preserves the norm automatically also preserves the dot product. This is a standard result for inner product spaces.
- edflsafoiewq 8y agoAh, polarization? And for it preserving the norm you can use the conjugate. Easy! Thanks!
- xelxebar 8y agoA little more exotic, but if you look here https://en.m.wikipedia.org/wiki/Plate_trick https://en.m.wikipedia.org/wiki/Plate_trick you can read about some simple rotating systems that, in some sense, end up reversed after a 360 degree rotation amd require another 360 to turn fully around. The article above has a nifty gif of this. Formally, this kind of thing is studied using what are called spinors and gets used a lot in quantum mechanics and friends when talking about quantum spin. Behind the scenes these are described using things called spinors. As it turns out, quaternions (and octonions) are capable of describing such things while simple rotations of any kind are insufficient. There are deep connections between spinors of different dimensions and the quaternions/octonions. For those interested, the term here is "Bott periodicity".
- chombier 8y agoYou just have to show that the quaternion norm is multiplicative, which follows easily from its definition |q|^2 = q * conj(q).