The conjugate can be used to swap the relative frames described by an orientation. For example, if \( {}^BQ_A \) describes the orientation of frame \( B \) relative to frame \( A \), the conjugate of \( \overline { {}^BQ_A }={}^AQ_B \) describes the orientation of frame \( A \) relative to frame \( B \). Relative orientations can also be seen as transformation from one frame to another.

The quaternion conjugate is denoted by one of the following notation: \( Q^* \), \( \overline Q \) or \(Q^T \). The prefered notation is \( \overline Q \).

Considere the quaternion \( Q \) defined by:

$$ Q = \left[ \begin{matrix} a && b && c && d \end{matrix} \right] $$

The conjugate of \( Q \) is given by:

$$ \overline Q = \left[ \begin{matrix} a && -b && -c && -d \end{matrix} \right] $$

- Quaternions and gyroscope
- Quaternion normalization
- Quaternion product
- Quaternion to rotation matrix
- Quaternions and rotations

Last update : 11/02/2022