Hadamard Matrices

H·Hᵀ = n·I — rows are mutually orthogonal ±1 vectors

Sylvester construction: H_{2n} = H₂ ⊗ H_n = [[H_n, H_n],[H_n, −H_n]]

Hadamard matrices have all entries ±1 and mutually orthogonal rows. The Sylvester (Kronecker product) construction gives H of order 2^k. They are related to Walsh functions, binary Hamming codes, and optimally efficient error-correcting codes. Hadamard conjecture: exists for all n ≡ 0 (mod 4).