Quaternion Conjugate

Quaternion conjugation reverses the sign of the vector part of a quaternion while leaving the scalar part unchanged. If

then the quaternion conjugate is

(1)

Conjugation is one of the central operations in quaternion algebra. It separates scalar and vector parts, produces a real quantity when a quaternion is multiplied by its conjugate, reverses the order of a product, and becomes the multiplicative inverse when the quaternion has unit norm.

In the later PhysicsLibrary attitude articles, unit quaternions represent passive coordinate transformations. Under that convention, conjugation has a direct frame meaning: it reverses the direction of the coordinate map.

1 Definition

For

define

(2)

In scalar first component form,

(3)

Thus conjugation changes the sign of the three imaginary coefficients and leaves the real coefficient unchanged.

The notation q^∗ is used throughout the PhysicsLibrary quaternion series. Other sources may use q, q^†, or another symbol. A dagger is also widely used for conjugate transpose of matrices, so an unfamiliar source should always be checked before importing formulas.

2 Basic examples

If

then

(4)

If q is real,

then

(5)

If q is pure,

then

(6)

The real quaternions are therefore exactly the quaternions fixed by conjugation, while pure quaternions reverse sign.

3 Conjugation is an involution

Applying conjugation twice returns the original quaternion:

(7)

Indeed,

gives

and a second conjugation gives

An operation that is its own inverse is called an involution.

4 Conjugation and addition

Let

Then

Conjugating gives

(8)

Similarly, for a real scalar a,

(9)

Thus conjugation is linear over real scalars.

5 Conjugation reverses the order of multiplication

The most important structural identity is

(10)

The factor order reverses. This is essential because quaternion multiplication is not commutative.

To derive the result, write

The Hamilton scalar vector product is

Conjugating this product changes the sign of its vector part:

Now

Their product is

Because the dot product is symmetric and

this becomes exactly

Therefore

Conjugation is consequently an anti homomorphism of quaternion multiplication: it preserves multiplication only after reversing factor order.

6 Conjugating a product of many factors

Repeated use of equation (10) gives

(11)

Every factor is conjugated and the entire order is reversed.

For three factors,

This identity becomes especially useful in frame composition and in simplifying quaternion coordinate transformation expressions.

7 Scalar and vector parts from conjugation

Conjugation gives convenient formulas for extracting the scalar and vector parts of a quaternion.

Add q and q^∗:

Therefore

(12)

Subtract:

Therefore

(13)

These formulas characterize the decomposition of a quaternion into parts that are unchanged and sign reversed by conjugation.

8 Product with the conjugate

Let

Using the scalar vector product,

The scalar part is

and the vector part is

Hence

(14)

The result is always a nonnegative real number.

This identity is the basis for the quaternion norm. It also leads directly to the inverse of every nonzero quaternion.

9 Relation to norm and inverse

The quaternion norm is defined by

Therefore

(15)

If q≠0, divide by ∥q∥² to obtain

(16)

For a unit quaternion,

so

(17)

This special case is fundamental in attitude mechanics because orientation quaternions are maintained at unit norm.

10 Conjugation of a passive frame quaternion

The algebraic operation above does not depend on any rotation convention. Its physical interpretation does.

PhysicsLibrary uses a unit quaternion

to map coordinates from frame A into frame B:

(18)

Because the quaternion is unit,

Therefore the inverse coordinate map is

(19)

It follows that

(20)

Conjugating a unit frame quaternion therefore reverses the map direction:

becomes

This is the quaternion counterpart of transposing a passive direction cosine matrix.

11 Passive axis angle interpretation

Suppose frame B is obtained from frame A by a positive right hand rule frame rotation through angle 𝜃 about unit axis u. Under the PhysicsLibrary passive convention,

(21)

Conjugation gives

(22)

Thus conjugation reverses the passive frame transformation. The same numerical quaternion on the right side of equation (23) is also the familiar positive active rotor for the corresponding physical vector rotation in a fixed frame.

This distinction is why a sign change in the vector part must not be described without also stating the intended frame map.

12 Relationship to direction cosine matrices

Let

be the passive direction cosine matrix corresponding to the unit quaternion ^Bq _A.

The inverse frame map satisfies

Since a direction cosine matrix is orthogonal,

Quaternion conjugation gives the same result:

(23)

Thus, for unit frame quaternions,

(24)

This provides a useful implementation check between quaternion and matrix orientation code.

13 Conjugation and frame composition

Suppose

maps A coordinates into B, and

maps B coordinates into C.

The direct passive frame map is

(25)

Conjugate both sides:

Using the inverse frame labels,

(26)

The reversed product order produced by conjugation is exactly what is required when the entire frame chain is traversed in the opposite direction.

14 Worked example 1: numerical conjugate

Let

Then

(27)

Applying conjugation again returns

15 Worked example 2: product with the conjugate

For

equation (15) gives

Therefore

(28)

The result is purely real.

The norm is consequently

and the inverse is

16 Worked example 3: order reversal

Let

First,

Therefore

Now compute the reversed conjugate product:

Expanding,

Hence

If one incorrectly used p^∗q^∗, the sign of the k term would be wrong.

17 Worked example 4: reversing a passive frame rotation

Let frame B be obtained from frame A by a positive 90^∘ rotation about +z. The PhysicsLibrary passive quaternion is

Its conjugate is

The first quaternion maps coordinate components from A to B. The conjugate maps them from B back to A.

If a fixed physical vector has

then the positive frame rotation produces

Applying the conjugate frame map returns

18 Common pitfalls

1. Changing the scalar part. Quaternion conjugation leaves q_w unchanged and reverses only the vector part.
2. Forgetting product order reversal. The correct identity is
   

   not p^∗q^∗.

3. Assuming q^∗ = q⁻¹ for every quaternion. This is true only for unit quaternions. In general,
   
4. Confusing quaternion conjugation with component storage order. Scalar first versus scalar last storage does not change the definition of the conjugate.
5. Confusing quaternion conjugation with complex conjugation of a matrix. The operations are related in some representations but are not the same object.
6. Conjugating a passive frame quaternion without reversing its frame meaning. For unit frame quaternions,
   
7. Changing Hamilton multiplication because the attitude interpretation is passive. Conjugation and passive frame interpretation do not change the Hamilton product.

19 Exercises

The exercises are stated first so the article can be used for self study. Complete solutions follow afterward.

1. Direct conjugation. For
   

   find q^∗.

2. Pure and real cases. Find the conjugate of
   

   and of

   
3. Double conjugation. Prove directly that
   
4. Scalar and vector extraction. Starting from q = q_w + q, derive
   

   and

   
5. Product with the conjugate. For
   

   compute qq^∗.

6. Order reversal. Let
   

   Compute (pq)^∗ and q^∗p^∗ and verify that they agree.

7. Incorrect order. For the same p and q, compute p^∗q^∗ and show that it differs from (pq)^∗.
8. Unit quaternion inverse. If ∥q∥ = 1, prove that q^∗ = q⁻¹.
9. Passive frame inverse. If ^Bq _A is a unit quaternion mapping A coordinates into B, write the quaternion that maps B coordinates back into A.
10. Frame chain reversal. Given
    

    conjugate the equation and derive the inverse frame chain.

20 Solutions

1. Direct conjugation

Starting from

reverse all three imaginary coefficients:

2. Pure and real cases

A real quaternion is unchanged:

For the pure quaternion

the conjugate is

3. Double conjugation

If

then

Conjugating again gives

4. Scalar and vector extraction

Add:

Therefore

Subtract:

Therefore

5. Product with the conjugate

For

the squared norm is

Hence

6. Order reversal

First,

Thus

Now

Then

Therefore

7. Incorrect order

Compute

This differs from

The factor order must reverse.

8. Unit quaternion inverse

From

and ∥q∥ = 1,

Similarly,

Therefore q^∗ is both the left and right inverse of q:

9. Passive frame inverse

The inverse coordinate map is represented by the conjugate:

10. Frame chain reversal

Begin with

Conjugating gives

Relabeling each conjugate as the inverse frame map,

Thus the conjugated chain traverses the same sequence of frames in the reverse direction.

21 What comes next

Quaternion conjugation immediately produces the real quantity

The next PhysicsLibrary article develops the quaternion norm from this identity and proves its important multiplicative properties. The following inverse article then uses conjugation and norm together to obtain

Later orientation articles use the unit quaternion specialization q⁻¹ = q^∗ to reverse passive frame transformations, invert direction cosine matrices, and derive coordinate transformation and attitude kinematic formulas.

22 Sources and historical notes

Quaternion conjugation appears naturally in Hamilton’s algebra and in later quaternion texts under several historical notations. Modern engineering notation often writes q^∗ or q and interprets conjugation as reversing the vector part.

Hamilton’s Elements of Quaternions and Joly’s A Manual of Quaternions provide public domain historical treatments. The modern PhysicsLibrary presentation emphasizes the anti order property, the relationship to norm and inverse, and the passive frame interpretation needed in mechanics and navigation.

References

License

Unless otherwise noted, this PhysicsLibrary entry is intended for release under the Creative Commons Attribution ShareAlike 4.0 International license.