= Skew-symmetric Householder tridiagonalization
{title2=$A'=A+vw^T-wv^T,\quad w=2Av/(v^Tv)$}
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 $v^TAv$ 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.
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