Quaternion conjugate

Introduction

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.

Notation

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

Properties

Considere the quaternion \( Q \) defined by:

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

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

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

See also

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

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