Multiplying by quaternions on the left and right

The map that takes a quaternionx to the quaternion qx is linear, so it can be represented as multiplication by a matrix. The same is true of the map that takesx toxq, but the two matrices are not the same because quaternion multiplication does not commute.

Letq =a + bi + cj +dkand let _qM be the matrix that represents multiplication on the left by q. Then

Now let M_q be the matrix that represents multiplication on the right by q. Then

Can prove both matrix representations are correct by showing that they do the right thing whenq = 1,i,j, andk. The rest follows by linearity.

You might speculate that the matrix representation for multiplying on the right by q might be the transpose of the matrix representation for multiplying on the left byq. You can look at the matrices above and see that's not the case.

In this post I talk about how to represent rotations with quaterions, and in this post I give an equation for the equivalent rotation matrix for a rotation described by a quaternion. You can prove that the matrix representation is correct by multiplying out _qM and M_q* . Keep in mind thatq in that case is a unit quaterion, so the sum of the squares of its components add to 1.

Related posts

• Dot, cross, and quaternion Products
• Faster quaternion rotations
• Matrix representation of number systems

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