Constructing the [v]ₓ Matrix from Skew-Symmetry and the Null Space
Let v = 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, K = 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 ...