Constructing the [v]ₓ Matrix from Skew-Symmetry and the Null Space
Let
| p |
| q |
| r |
| 0 |
| 0 |
| 0 |
The objective is to construct the standard 3 × 3 matrix [v]ₓ associated with v, without assuming its entries in advance. Begin with two requirements:
Kᵀ = −K
Kv = 0.
The first condition requires skew-symmetry. The second requires v to lie in the null space of K.
1. Write the general skew-symmetric matrix
Since Kᵀ = −K, the diagonal entries must be zero and entries reflected across the diagonal must have opposite signs. Therefore,
| 0 | α | β |
| −α | 0 | γ |
| −β | −γ | 0 |
At this stage, α, β and γ are unknown.
2. Require Kv = 0
| 0 | α | β |
| −α | 0 | γ |
| −β | −γ | 0 |
| p |
| q |
| r |
| 0 |
| 0 |
| 0 |
Multiplication gives
| αq + βr |
| −αp + γr |
| −βp − γq |
| 0 |
| 0 |
| 0 |
Hence the unknown entries must satisfy
3. Find and verify a solution
Choose
α = −r, β = q, γ = −p.
Verify the first equation:
αq + βr
= (−r)q + qr
= −rq + qr
= 0.
Verify the second equation:
−αp + γr
= −(−r)p + (−p)r
= rp − pr
= 0.
Verify the third equation:
−βp − γq
= −qp − (−p)q
= −qp + pq
= 0.
Thus the chosen values satisfy all three equations.
4. Substitute into K
| 0 | −r | q |
| r | 0 | −p |
| −q | p | 0 |
The standard notation for this matrix is
| 0 | −r | q |
| r | 0 | −p |
| −q | p | 0 |
The conditions Kᵀ = −K and Kv = 0 allow scalar multiples of this matrix. The displayed matrix selects the standard scale and orientation.
5. Final verification
Transposing the matrix gives
| 0 | r | −q |
| −r | 0 | p |
| q | −p | 0 |
Therefore, [v]ₓ is skew-symmetric. Applying it to v gives
| −rq + qr |
| rp − pr |
| −qp + pq |
| 0 |
| 0 |
| 0 |
Hence the construction is complete:
| 0 | −r | q |
| r | 0 | −p |
| −q | p | 0 |
[v]ₓᵀ = −[v]ₓ, [v]ₓv = 0.
