Skew-symmetric Householder tridiagonalization ($A^{\prime }=A+v{w}^T-w{v}^T,w=2Av/(v^{T}v)$)

An orthogonal similarity preserves skew symmetry, so upper-Hessenberg reduction of a real skew-symmetric matrix is automatically tridiagonal with zero diagonal. For a Householder reflection, the scalar $v^{T}Av$ vanishes, eliminating the fourth term of the general similarity update. Storing only one triangle and using the displayed skew rank-two update costs $2n^3/3+O(n^2)$ scalar products, the same leading product count as optimized symmetric tridiagonalization; the simplification saves lower-order work.