Constructing the General 3 × 3 Skew-Symmetric Matrix from First Principles
A skew-symmetric matrix is a square matrix whose transpose equals its negative. The construction below begins with an arbitrary 3 × 3 matrix, subtracts its transpose, and derives the complete general form without assuming the result in advance. K is skew-symmetric precisely when Kᵀ = −K. 1. Begin with an arbitrary matrix Let A = a b c d e f g h i and Aᵀ = a d g b e h c f i Transposition reflects the entries across the main diagonal: rows become columns and columns become rows. 2. Subtract the transpose Define K = A − Aᵀ. Subtract corresponding entries: K = a−a b−d c−g d−b e−e f−h g−c h−f i−i The diagonal entries cancel: ...
